GEOMETRY OF THE ENDS OF THE MODULI SPACE OF ANTI-SELF-DUAL CONNECTIONS

Paul M. N. Feehan

1. Introduction

Let X_{0} be a closed, oriented, C^{∞} four-manifold and let M_{X_{0},P} (g_{0}) be the moduli space of g_{0}-anti-self-dual connections on a principal G bundle P over X_{0}. The subspace M_{X_{0},P}^{∗} (g_{0}), obtained by excluding the reducible connections is then a finite-dimensional, usually non-compact, C^{∞} manifold. The moduli space M_{X_{0},P}^{∗} (g_{0}) is naturally endowed with a metric g of Weil-Petersson type, called the L^{2} metric, and our purpose in this article is to study the geometry of the moduli space ends.

(a) Main results. It has been conjectured by D. Groisser and T. Parker in [G-P87], [G-P89] and by S. K. Donaldson in [D90a] that the moduli space of anti-self-dual connections, endowed with the L^{2} metric, has finite volume and diameter. The goal of this article is to prove this conjecture under the hypotheses described below.

Theorem 1.1. Let X_{0} be a closed, connected, oriented, simply-connected, C^{∞} fourmanifold with generic metric g_{0} and let P be a principal G bundle over X_{0} such that either (1) G = SU(2) or SO(3) and b^{+}(X_{0}) = 0, or (2) G = SO(3) and w_{2}(P) ≠ 0, where w_{2}(P) is the second Stiefel-Whitney class of P. Then the moduli space M_{X_{0},P}^{∗} (g_{0}) of irreducible g_{0}-anti-self-dual connections on P has finite volume and diameter with respect to the L^{2} metric g defined by g_{0}.

We plan to discuss the case of G = SU(2) and b^{+}(X_{0}) > 0 in a subsequent article. Note that when G = SO(3) and w_{2}(P) ≠ 0, the trivial (product) connection Θ does not appear in the Uhlenbeck compactification \overline{M}_{X,P} (g_{0}). By ‘diameter’ we mean the sum of the diameters of the connected components of M_{X_{0},P}^{∗} (g_{0}); the hypotheses imply that M_{X_{0},P}^{∗} (g_{0}) has finitely many path components. In [D89] Donaldson conjectured that the L^{2}-metric completion of the moduli space coincides with the Uhlenbeck compactification [D86], [D-K]. We announce here the following result whose proof is included in [F94].

Theorem 1.2. Under the hypotheses of Theorem 1.1, the completion of M_{X_{0},P}^{∗} (g_{0}) with respect to the L^{2} metric g is homeomorphic to the Uhlenbeck compactification \overline{M}_{X_{0},P}^{u} (g_{0}).

The requirement that X_{0} be simply-connected implies that the moduli space of flat connections consists of a single point representing the product connection over X_{0}. This assumption simplifies the description of the ends of the moduli spaces M_{X_{0},P}^{∗} (g_{0}), but is not important in the derivation of bounds for the components of g. We assume G = SU(2) or SO(3) in order to appeal to the generic metric theorems of Freed and Uhlenbeck which ensure that the moduli space is a C^{∞} manifold: otherwise, the bounds for g obtained in Chapter 5 hold for any compact Lie group. For the sake of clarity, we assume G = SU(2) for the remainder of the article and denote M_{X_{0},P} (g_{0}) by M_{X_{0},k}(g_{0}), where c_{2}(P) = k ≥ 0 is the second Chern class.

Research supported in part by an NSF Mathematical Sciences Postdoctoral Fellowship under grant DMS 9306061.

(b) History. The properties of the L^{2} metric have been investigated by many authors in recent years, but most extensively by Groisser and Parker. In particular, they have conducted detailed studies of its behaviour at the boundary of certain k = 1 moduli spaces. Explicit formulas for the components of g have been found by Doi, Matsumoto, and Matumoto [D-M-M], Groisser and Parker [G-P87], and Habermann [Hab] when k = 1 and X_{0} is the four-sphere 𝕊^{4} with its standard round metric g_{1}. Groisser conducted a similar study when X_{0} is the complex projective space \overline{ℂℙ}^{2}, equipped with the Fubini-Study metric g_{FS} [G90]. Their formulas imply that these k = 1 moduli spaces have finite g-volume and g-diameter. More generally, Groisser and Parker have established Theorem 1.1 in the special case k = 1 [G-P89]. They also obtained C^{0} bounds for g in neighbourhoods of the reducible connections, the ‘conical ends’, for any k ≥ 1. In [G92], Groisser refined some of the k = 1 results obtained in [G-P89]. It is worth recalling that the L^{2} metric is not invariant with respect to conformal changes in the metric g_{0} on X_{0}.

The approach of [G-P89] does not appear to readily generalise to the case k > 1, since their method relies on Donaldson’s collar map which gives a diffeomorphism from the ‘bubbling end’ of M_{X_{0},1}^{∗}(g_{0}) to the collar X_{0} ×(0, λ_{0}). For this reason we adopt a quite different method which uses the gluing techniques of Taubes and Donaldson to construct a system of local coordinate charts covering the ‘ends’ of the moduli space. We then estimate the components of g with respect to these coordinates. In the case of the Weil-Petersson metric on Teichmüller space, estimates of this type have been obtained by Masur [Mas]. In [F92], the author proved Theorem 1.1, when X_{0} = 𝕊^{4} and k = 2, using the ADHM correspondence [D-K]. After the present work was submitted, a preprint was received from Peng giving L^{2} estimates for the derivatives with respect to moduli parameters of the family of anti-self-dual connections A on the connected sum X_{0}#_{λ}𝕊^{4} constructed in §7.2.2 of [D-K], with H_{A_{i}}^{2} = 0 [Pe]. His L^{2} estimates are defined with respect to a family of metrics g_{λ} which are conformally equivalent to g_{0} and pinch the neck of the connected sum as λ → 0, away from the neck coinciding with g_{0} on X_{0} and converging in C^{2} to the standard round metric on the unit sphere 𝕊^{4}.

(c) Outline and strategy. It remains to summarise the methods used in the proofs of our main results. Let us first recall the definition of the L^{2} metric. The tangent space T_{A}M_{X_{0},k}^{∗}(g_{0}) is identified with the cohomology group H_{A}^{1} = ker d_{A}^{+,g_{0}}/im d_{A}^{∗,g_{0}}. Given tangent vectors [a], [b], the L^{2} metric g is defined by

(1.1)  g_{[A]}([a], [b]) = (π_{A}a, π_{A}b)_{L^{2}(X_{0},g_{0})},

where π_{A} = 1 − d_{A}(d_{A}^{∗,g_{0}}d_{A})^{−1}d_{A}^{∗,g_{0}} is the L^{2} orthogonal projection from L^{2}Ω^{1}(X_{0}, ad P) to the subspace ker d_{A}^{∗,g_{0}}. Clearly, g([a], [b]) is bounded above by ∥a∥_{L^{2}}∥b∥_{L^{2}}, and so a reasonable strategy is to seek upper bounds for g over the moduli space ends. This will suffice for our present application.

(i) Moduli space ends and the bubble tree compactification. Our first task is to describe useful models for the ends of the moduli space of anti-self-dual connections. Let (A_{0}, x_{1}, …, x_{m_{0}}) be a point in the stratum \overline{M}_{X_{0},k}^{u}(g_{0}) ∩ (M_{X_{0},k_{0}}(g_{0}) × s^{k−k_{0}}(X_{0})) of the Uhlenbeck compactification (see §4.1) which lies away from the diagonals of the symmetric product, so that m_{0} = k − k_{0} and each point x_{i} has multiplicity 1. Then every point [A] ∈ M_{X_{0},k}(g_{0}) which is close enough to (A_{0}, x_{1}, …, x_{m_{0}}) in the Uhlenbeck topology can be shown to lie in a neighbourhood constructible by gluing or ‘gluing neighbourhood’ [D86], [D-K]. Thus, suppose [A_{α}] is a sequence in M_{X_{0},k}(g_{0}) which converges weakly to (A_{0}, x_{1}, …, x_{m_{0}}). As described in §4.2, the sequence of connections [A_{α}] produces sequences of local mass centres x_{iα} converging to the points x_{i} and sequences of local scales λ_{iα} converging to zero. Using the scales λ_{iα}, one now dilates the metric g_{0} around the points x_{iα} and produces a sequence of conformally equivalent, C^{∞} metrics g_{α} on a connected sum X ≡ X_{0}#_{i=1}^{m_{0}}𝕊^{4}. As the scales λ_{iα} tend to zero, the corresponding neck is pinched and the connected-sum metrics g_{α} converge in C^{∞} on compact subsets away from the neck regions to the metric g_{0} on X_{0} and the standard round metric g_{1} (of radius 1) on each copy of 𝕊^{4}. This ‘conformal blow-up’ procedure gives a sequence of g_{α}-anti-self-dual connections [Ǎ_{α}] which converges strongly (in the sense of [D-K]) to a limit (A_{0}, I_{1}, …, I_{m_{0}}) over the join X_{0} ∨_{i=1}^{m_{0}} 𝕊^{4}, where the I_{i} are the standard one-instantons over X_{i} = 𝕊^{4} with centre at the north pole n and scale 1. Here, strong convergence means C^{∞} convergence on compact sets away from the necks and such that c_{2}(A_{0}) + ∑_{i=1}^{m_{0}} c_{2}(I_{i}) = k: there are no singular points and there is no curvature loss over the necks. One obtains an open neighbourhood in \overline{M}_{X_{0},k_{0}}^{u}(g_{0}) of the boundary point (A_{0}, x_{1}, …, x_{m_{0}}) by gluing up the limit (A_{0}, I_{1}, …, I_{m_{0}}).

On the other hand, if the set Z_{0} ≡ (x_{1}, …, x_{m_{0}}) lies in the diagonal of the symmetric product s^{k−k_{0}}(X_{0}), the limiting behaviour of the sequence [A_{α}] may be rather more complicated. Suppose [Ǎ_{α}] is the corresponding sequence of g_{α}-anti-self-dual connections over X = X_{0}#_{i=1}^{m_{0}}𝕊^{4} produced by conformal blow-ups. The sequence Ǎ_{α} converges in C^{∞} on compact subsets of X_{0} \ Z_{0} to a g_{0}-ASD connection A_{0} over X_{0}, but in general only converges weakly to an Uhlenbeck limit (A_{i}, Z_{i}) over the four-spheres X_{i} ≡ 𝕊^{4}, where Z_{i} = (x_{i1}, …, x_{im_{i}}) is contained in X_{i} \ {s} and s is the south pole. If the connection A_{i}, i > 0, is not flat, then the conformal blow-ups may be chosen so that it is centred in the sense of [T88]: its mass centre lies at the north pole and it has scale (essentially its ‘standard deviation’) equal to 1 (see §4.2).

Unless all the singular sets Z_{i} are empty, one can no longer produce an open subset of the moduli space M_{X_{0},k}(g_{0}) simply by gluing up the connections (A_{i})_{i=0}^{m_{0}}: because of the nature of the convergence process, some of the required moduli parameters have been lost in the limit.

Instead, the conformal blow-up process above must be iterated. The idea of iterating conformal blow-ups has been suggested by Sacks and Uhlenbeck in the context of harmonic maps of 𝕊^{2} [S-U]. Taubes described an iterative scheme of this type which is used to analyse the limiting behaviour of sequences of connections with uniformly bounded Yang-Mills functional and functional gradient tending to zero [T88]. Parker and Wolfson described a bubble tree compactification for pseudoholomorphic maps of Riemann surfaces into symplectic manifolds and noted that their method should apply to the case of Yang-Mills connections over four-manifolds [P-W].

For the problem at hand, by repeatedly applying conformal blow-ups, we obtain a sequence of g_{α}-anti-self-dual connections Ǎ_{α} over a large connected sum X ≡ #_{I∈ℐ}X_{I}. Here, ℐ is a set of multi-indices I obtained when the conformal blow-up process is iterated. Thus, ℐ records the tree structure and if I = 0, then X_{I} is the four-manifold X_{0}, while if I ≠ 0, then X_{I} is a copy of 𝕊^{4}. The construction of the ‘conformal blow-up maps’ f_{Iα} ensures that the blow-up process must be repeated at most k times in order to produce a sequence of connections [Ǎ_{α}] which converge strongly to a limit (A_{I})_{I∈ℐ} over a join ∨_{I∈ℐ}X_{I}, where A_{0} is a g_{0}-anti-self-dual connection over X_{0} and each A_{I}, for I ≠ 0, is a g_{1}-anti-self-dual connection over X_{I} = 𝕊^{4}. The sequence of metrics g_{α} converges in C^{∞} on compact subsets away from the neck regions to the metric g_{0} on X_{0} and the standard round metric g_{1} on each sphere X_{I}. This convergence scheme produces the ‘bubble tree compactification’ \overline{M}_{X_{0},k_{0}}^{τ}(g_{0}) and is described in §4.3.

In particular, bubble tree degeneration and gluing are inverse to one another in a natural way. One can now glue up the bubble tree limits (A_{I})_{I∈ℐ} to form g-anti-self-dual connections A over a connected sum X ≡ #_{I∈ℐ}X_{I} using the techniques of [D-K] and construct open subsets of the moduli space M_{X,k}(g) by small deformations of the limit data. The gluing procedure gives a collection of conformal maps f_{I} (from a small ball in a lower level summand X_{I_{−}} to the complement in the sphere X_{I} of a small ball around the south pole) defined in exactly the same way as the conformal blow-up maps f_{Iα} above. Here, g is a C^{∞} metric on X which is conformally equivalent to the old metric g_{0} (via the maps f_{I}) and depends on the choice of gluing sites, frames in the principle SO(4) frame bundle F X_{0}, scales, and the metric g_{0} on X_{0}: its construction and properties are discussed in §3.5. Similar metrics over connected sums are described in [D86] and [T92]. Pulling back via the blow-up maps then gives g_{0}-anti-self-dual connections Â over X_{0} and hence, produces open subsets of the moduli space M_{X_{0},k}(g_{0}).

Generalising the arguments in [D86] and [D-K] and employing the compactness results of §4.3, one then shows that \overline{M}_{X_{0},k}^{u}(g_{0}) has a finite cover consisting of gluing neighbourhoods \overline{𝒱}. Of course, any precompact open subset of M_{X_{0},k}(g_{0}) is covered by finitely many Kuranishi charts and these comprise the ‘gluing charts’ in this case. Moreover, the L^{2}-metric geometry near the reducible connections, the conical ends, has already been analysed by Groisser and Parker [G-P89], so we may confine our attention to the more troublesome bubbling ends.

(ii) Upper bounds for the components of the L^{2} metric. We now outline a method of computing estimates for the L^{2} metric g over the ends of the moduli space. In §§3.3 and 3.4 we apply the techniques of [D86] and [D-K] to first construct approximate gluing maps 𝒥′ : 𝒯^{0}/Γ → B_{X,k}^{∗}, t → [A′(t)]. Here, X is the connected sum #_{I∈ℐ}X_{I} with C^{∞} metric g conformally equivalent to g_{0} on X_{0}, and 𝒯^{0}/Γ is a certain parameter space. If the gself-dual curvature F^{+,g}(A′) is sufficiently small one can then solve the g-anti-self-duality equation F^{+,g}(A′ + a) = 0, or equivalently

(1.2)  d_{A′}^{+,g}a + (a ∧ a)^{+,g} = −F^{+,g}(A′),

for a ∈ Ω^{1}(X, ad P). This gives a C^{∞} family of g-anti-self-dual connections A ≡ A′ + a and thus a gluing map 𝒥 : 𝒯^{0}/Γ → M_{X,k}^{∗}(g), t → [A(t)]. The solutions a to Eq. (1.2) are expressed in the form a = P ξ, where ξ ∈ Ω^{+,g}(X, ad P) and P is a right inverse to the operator d_{A′}^{+,g} (constructed as in [D-K] by patching together right inverses P_{I} for the operators d_{A_{I}}^{+,g_{I}} over the summands X_{I}. Therefore, Eq. (1.2) takes the shape

(1.3)  ξ + (P ξ ∧ P ξ)^{+,g} = −F^{+,g}(A′).

Following [D-K], we assemble the framework required for solving Eq. (1.3) in §5.1.

Now the L^{2} metric g depends on the choice of metric g_{0}, not just the conformal class [g_{0}]. So, using the conformal maps f_{I}, we pull back the family of g-anti-self-dual connections A(t) = A′(t) + a(t) over X to an equivalent C^{∞} family of g_{0}-anti-self-dual connections Â(t) = Â′(t) + â(t) over X_{0}. Hence, we obtain gluing maps 𝒥̂ : 𝒯^{0}/Γ → M_{X_{0},k}^{∗}(g_{0}), t → [Â(t)] analogous to those constructed by Taubes. The properties of the gluing maps 𝒥 and 𝒥̂ are discussed in §5.2.

The problem then is to estimate the differentials D𝒥̂ and this task is comprised of two parts. The first part is to bound the derivatives ∂A′/∂t: this local calculation is the subject of §§3.7 to 3.9 and the main results are summarised in §3.10. The more difficult part is to bound the derivatives of the correction terms, ∂â/∂t: this involves bounding the derivatives of global operators such as P and is described in §§5.3 to 5.5. The problem of expressing bounds for derivatives of â(t) in terms of bounds for derivatives of a(t) is the subject of §3.5. Some care is required here, since the conformal maps f_{I} vary with the scale and centre parameters, as does the metric g in Eq. (1.3). The required estimates for the derivatives ∂a/∂t are then computed in §§5.3 to 5.5 in terms of bounds for ∂P/∂t and ∂ξ/∂t; the estimates for ∂ξ/∂t are obtained implicitly from Eq. (1.3). For the special case of a neighbourhood of a point (A_{0}, A_{1}) (with H_{A_{0}}^{2} = 0), L^{2} estimates for the derivatives ∂A/∂t were later obtained independently by Peng using similar methods [Pe].

It is the estimates for derivatives with respect to the scales λ_{I} which require the most care. For example, difficulties arise when bounding the derivatives ∂Â′/∂λ_{I} because of the dependence on λ_{I} of the conformal maps f_{I} and the cut-off functions required to patch the connections A_{I} together over the connected sum. These derivatives are ill-behaved as λ_{I} → 0 and the necks of the connected sum X are pinched. Problems also occur when one attempts to bound ∂a/∂λ_{I}, since a = P ξ and the construction of P involves cut-off functions with badly behaved derivatives with respect to λ_{I} as λ_{I} → 0. The final estimates for the differentials D𝒥̂ and the corresponding bounds for the L^{2} metric g are sumarised in §5.6. The constants appearing in the bounds for g depend only on the gluing neighbourhood. Theorem 1.1 then follows immediately from these estimates.

Acknowledgements. The author would like to thank J. W. Morgan, T. Mrowka, D. H. Phong, and C. H. Taubes for helpful conversations. The current work was begun at Columbia University and completed while the author was a postdoctoral fellow at the Mathematical Sciences Research Institute in 1992–93. He would like to thank these institutions and the Department of Mathematics at Harvard University for their generous support and hospitality.

2. Preliminaries

In this Chapter we establish our notation and define the L^{2} metric. Unless stated otherwise, we adhere to the standard conventions of [D-K]. For further details concerning gauge theory, we refer to [D-K] or [F-U] and the references therein, while for details concerning the L^{2} metric, we refer to [G-P87], [G-P89].

Let X be a closed, connected, oriented, C^{∞} four-manifold with Riemannian metric g and let P be a principal G bundle over X with Lie algebra 𝔤. As noted in the Introduction, we will generally confine our attention in this article to the case G = SU(2) for the sake of clarity. We let Ω^{l}(P, 𝔤) denote the space of C^{∞} 𝔤-valued l-forms, let ad P = P ×_{Ad} 𝔤 be the adjoint bundle, and let Ω^{l}(X, ad P) be the space of C^{∞} ad P -valued l-forms on X. Let A_{P} be the affine subspace in Ω^{1}(P, 𝔤) of C^{∞} connection 1-forms on P. For a connection A on P, we let ∇_{A} be the corresponding covariant derivative, let d_{A} be the exterior covariant derivative, and let F_{A} ∈ Ω^{2}(X, ad P) denote the curvature.

Let G_{P} be the group of C^{∞} bundle automorphisms or gauge transformations. Recall that the isotropy group Γ_{A} ⊂ G_{P} of a connection A on P is isomorphic to the centraliser of the holonomy group of A in G and the centre Z of the bundle structure group G is isomorphic to the centre of G_{P}. Thus Γ_{A} ⊃ Z and we let A_{P}^{∗} be the dense open subset of connections A ∈ A_{P} with Γ_{A} = Z, so that A_{P}^{∗} is the space of irreducible connections on P when G = SU(2) or SO(3).

The bundles Λ^{l}T^{∗}X ⊗ ad P have fibre metrics ⟨ , ⟩ induced by the Riemannian metric g on X and the inner product on the Lie algebra 𝔤 given by −1 times the Cartan-Killing form: if ξ_{1}, ξ_{2} ∈ 𝔤, then ⟨ξ_{1}, ξ_{2}⟩ = −tr(ξ_{1}ξ_{2}). In particular, we may define Sobolev spaces L_{n}^{p}Ω^{l}(X, ad P) in the usual way and consider the action of the L_{n−1}^{2} gauge transformations G on the space of L_{n}^{2} connections A_{P} (for n > 2) with quotient B_{P} = A_{P} /G_{P}, omitting the explicit Sobolev notation when no confusion can arise.

The tangent space T_{A}A_{P}^{∗} is equal to Ω^{1}(X, ad P) while the tangent space to the G-orbit through A ∈ A_{P}^{∗} is im d_{A} ⊂ Ω^{1}(X, ad P). This induces an L^{2}-orthogonal decomposition T_{A}A_{P}^{∗} = ker d_{A}^{∗} ⊕ im d_{A}, where ker d_{A}^{∗} ⊂ Ω^{1}(X, ad P). There is an associated horizontal projection operator π_{A} : T_{A}A_{P}^{∗} → ker d_{A}^{∗}, with π_{A} = 1−d_{A}G_{A}^{0}d_{A}^{∗}, where G_{A}^{0} is the Green’s operator for the Laplacian Δ_{A}^{0} = d_{A}^{∗}d_{A}. To identify the tangent space T_{[A]}B_{P}^{∗}, introduce C^{∞} paths A(t) in A_{P}^{∗} and u(t) in G_{P}, u(0) = 1. If A^{u}(t) ≡ u_{t}(A_{t}), then

(2.1)  \frac{dA^{u}}{dt} = Ad (u^{−1})\frac{dA}{dt} + d_{A^{u}} (u^{−1} \frac{du}{dt}) .

Thus dA/dt(0) defines an element of Ω^{1}(X, ad P)/im d_{A} and therefore the tangent space T_{[A]}B_{P}^{∗} is given by Ω^{1}(X, ad P)/im d_{A} ≃ ker d_{A}^{∗}.

Let M_{P} (g) be the moduli space of g-anti-self-dual connections on the G bundle P over X, that is {[A] ∈ B_{P} : F^{+,g}(A) = 0}, and let M_{P}^{∗} (g) be the dense open subset M_{P} (g)∩B_{P}^{∗}. If A(t) is a C^{∞} path in A_{P} satisfying F^{+,g}(A(t)) = 0, then dA/dt(0) defines an element of ker d_{A}^{∗}/im d_{A}^{+,g}. The g-anti-self-dual condition F^{+,g}(A) = 0 is equivalent to d_{A}^{+,g} ∘ d_{A} = 0 and so we have the elliptic deformation complex

(2.2)  Ω^{0}(X, ad P) ⟶^{d_{A}} Ω^{1}(X, ad P) ⟶^{d_{A}^{+,g}} Ω^{+,g}(X, ad P)

with associated cohomology groups H_{A}^{∗}, where H_{A}^{0} is the Lie algebra of Γ_{A}, the group H_{A}^{1} = ker d_{A}^{+,g}/im d_{A} is just the tangent space T_{[A]}M_{P} (g), and H_{A}^{2} = coker d_{A}^{+,g}. By Hodge theory there are natural isomorphisms H_{A}^{0} ≃ ker Δ_{A}^{0}, H_{A}^{1} ≃ ker d_{A}^{∗} ∩ ker d_{A}^{+,g}, and H_{A}^{2} ≃ ker Δ_{A}^{+,g}, where the Laplacian Δ_{A}^{+,g} is equal to d_{A}^{+,g}(d_{A}^{+,g})^{∗}.

If [A] is an irreducible point of M_{P} (g), then H_{A}^{0} = 0, and an irreducible point [A] is regular if H_{A}^{2} = 0. The moduli space M_{P} (g) is regular if all its irreducible points are regular points, and in that case, M_{P}^{∗} (g) is a C^{∞} manifold of dimension

(2.3)  dim M_{P} (g) = 8k(P) − 3(1 − b_{1}(X) + b^{+}(X)),

with tangent space T_{[A]}M_{P}^{∗} (g) = H_{A}^{1} at the point [A].

According to the Freed-Uhlenbeck theorems, the anti-self-dual moduli spaces M_{P}^{∗} (g) are smooth manifolds when g is generic. More precisely, if b^{+}(X) > 0, P is any SU(2) or SO(3) bundle P over X, and the metric g on X is generic, then (1) M_{P}^{∗} (g) contains no points [A] with H_{A}^{2} ≠ 0; (2) If b^{+}(X) > 0 and l > 0 then M_{P} (g) contains no points [A] with H_{A}^{0} ≠ 0 for any bundle P with 0 < k(P) ≤ l; (3) If b^{+}(X) = 0 and P is non-trivial, then the cohomology groups H_{A}^{2} are zero for all the reducible g-anti-self-dual connections A on P, and a neighbourhood of point [A] ∈ M_{P} (g) with H_{A}^{0} ≠ 0 is homeomorphic to a cone over ℂℙ^{4k−2} and diffeomorphic away from the cone point [A].

It remains to define the L^{2} metric. The quotient space B_{P}^{∗} inherits a (weak) Riemannian L^{2} metric g by requiring that the projection map for the principal G_{P} /Z bundle A_{P}^{∗} → B_{P}^{∗} be a Riemannian submersion: if [a], [b] are tangent vectors in T_{[A]}B_{P}^{∗}, then

(2.4)  g_{[A]}([a], [b]) ≡ ∫_{X}⟨π_{A}a, π_{A}b⟩ dV_{g},

and this restricts to give a C^{∞} Riemannian metric g on the moduli space M_{P}^{∗} (g).

3. Differentials of the approximate gluing maps

Our purpose in this Chapter is to construct the approximate gluing maps 𝒥′ : 𝒯/Γ → B_{X,k}^{∗} and 𝒥̂′ : 𝒯/Γ → B_{X_{0},k}^{∗}, and to estimate the differentials D𝒥′, and especially D𝒥̂′. The construction of 𝒥′ uses the method employed by Donaldson in [D86], [D-K]. The induced maps 𝒥̂′ are essentially the approximate gluing maps described by Taubes in [T82], [T84a], [T88]. In the former case, we obtain an almost g-anti-self-dual connection A′ over a connected sum X = X_{0}#_{I∈ℐ}𝕊^{4} with metric g conformally equivalent to g_{0} on X_{0}, while in the latter case we obtain an almost g_{0}-anti-self-dual connection Â′ over X_{0} with its fixed metric g_{0}. In Chapter 5, we obtain a system of coordinate charts 𝒥̂ : 𝒯/Γ → M_{X_{0},k}^{∗}(g_{0}) covering the moduli space by perturbing the maps 𝒥̂′ using the techniques of [D-K] for solving the anti-self-dual equation.

3.1. Preliminary estimates for connections and curvature. We describe some pointwise estimates for local connection one-forms and curvature two-forms. We first consider estimates for connection one-forms in radial gauge on a C^{∞} manifold X with C^{∞} metric g. Suppose P → X is a principal G bundle, A is a C^{∞} connection on P, and B is an open geodesic ball centred at x_{0} ∈ X with radius ϱ/2, where ϱ is the injectivity radius of (X, g). Define a C^{∞} local section σ : B → P by parallel transport of a point in the fibre P |_{x_{0}} along radial geodesics through x_{0}. If γ is a radial geodesic in B with γ(0) = x_{0} and γ̇(t) = ξ_{t}, then σ^{∗}A(x_{0}) = 0 and ι_{ξ_{t}}σ^{∗}A(γ(t)) = 0, t > 0. If ϕ^{−1} : B → ℝ^{4} is a geodesic normal coordinate system centred at x_{0} and we define a geodesic γ by γ(t) = ϕ(tx), x ∈ B, t ∈ [0, 1], then γ^{μ}(t) = tx^{μ}, γ̇ = x, and ι_{x}σ^{∗}A = x^{μ}(σ^{∗}A)_{μ}. We recall the following estimates for local connection one-forms in radial gauge.

Lemma 3.1. [U82, p. 14] Let A be a C^{∞} connection on a principal G bundle P → X, where X is a C^{∞} manifold with C^{∞} metric g, B is a geodesic ball of radius ϱ/2 centred at x_{0} ∈ X, σ : B → P is a local section such that σ^{∗}A is in radial gauge centred at x_{0}, and ϕ^{−1} : B → ℝ^{n} is a geodesic normal coordinate system centred at x_{0}. If K = ∥F_{A}∥_{L^{∞}(B,g)}, then |ϕ^{∗}σ^{∗}A|_{g}(x) ≤ K|x|, |x| < ϱ/2.

Let ℍℙ^{1} be the right quaternionic projective space, with the standard identifications ℍ ≃ ℝ^{4} and ℍℙ^{1} ≃ 𝕊^{4}. Coordinate patches for 𝕊^{4} may then be defined by U_{n} = {[x, y] : y ≠ 0} = 𝕊^{4} \{s} and U_{s} = {[x, y] : x ≠ 0} = 𝕊^{4} \{n} covering the north pole n = [0, 1] and south pole s = [1, 0], respectively. We let ϕ_{n}^{−1} : U_{n} → ℝ^{4}, [x, y] ↦ xy^{−1} and ϕ_{s}^{−1} : U_{s} → ℝ^{4}, [x, y] ↦ yx^{−1} denote the standard local coordinate charts. If g_{1} is the standard round metric of radius 1 on 𝕊^{4}, then

(3.1)  (ϕ_{α}^{∗}g_{1})_{μν}(x) = h_{1}^{2}(x)δ_{μν} = \frac{4}{(1 + |x|^{2})^{2}} δ_{μν},  x ∈ ℝ^{4},

for α = n, s; the standard flat metric on ℝ^{4} is denoted by δ.

Let A be a C^{∞} connection on a principal G bundle P → 𝕊^{4}, where 𝕊^{4} has its standard metric g_{1}. We define a system of local sections σ_{α} : U_{α} → P, α = n, s, by parallel transport of points in the fibres P |_{α} along radial geodesics through the north or south poles. The estimates below follow easily since A is smooth over 𝕊^{4} with metric g_{1}:

Lemma 3.2. Let A be a C^{∞} connection on a principal G bundle P → 𝕊^{4}, where 𝕊^{4} has metric g_{1} and K = ∥F_{A}∥_{L^{∞}(𝕊^{4},g_{1})}. Then, for α, β ∈ {n, s},

|ϕ_{β}^{∗}F (σ_{α}^{∗}A)|_{δ}(x) ≤ 4K \frac{1}{(1 + |x|^{2})^{2}} for \{ \begin{matrix}x ∈ ℝ^{4} & if α = β \\ x ∈ ℝ^{4} \ \{0\} & if α ≠ β. \end{matrix}

Lemma 3.3. Given the hypotheses of Lemma 3.2, if the local connection one-forms σ_{α}^{∗}A are in radial gauge, then |ϕ_{α}^{∗}σ_{α}^{∗}A|_{g_{1}}(x) ≤ K|x|, for x ∈ ℝ^{4} and α = n, s.

3.2. Connections over the four-sphere and conformal diffeomorphisms. Recall that the group of conformal diffeomorphisms of 𝕊^{4} acts on the space A_{P} of C^{∞} connections on a G bundle P over 𝕊^{4}. The group D × T of dilations and translations of ℝ^{4} may be identified with a subgroup of the conformal group of 𝕊^{4}. Hence, in this section we discuss some aspects of the induced action of ℝ^{+} × ℝ^{4} on the space A_{P}. For related material we refer to [D83], [F-U], [G-P87], [G-P89], and [T88].

Let P be a G bundle with C^{∞} connection A ∈ Ω^{1}(P, 𝔤) over a C^{∞} manifold X and suppose φ_{t} is a C^{∞} one-parameter group of diffeomorphisms of X generating a vector field ξ ∈ C^{∞}(T X). Let ξ̃ ∈ C^{∞}(T P) be the horizontal vector field covering ξ and let φ̃_{t} be the one-parameter group of diffeomorphisms of P generated by ξ̃. Then φ̃_{t} commutes with right G multiplication and covers φ. Fixing Ω ∈ Ω^{1}(P, 𝔤), we obtain a C^{∞} one-parameter family of C^{∞} one-forms φ̃_{t}^{∗}Ω on P with

(3.2)  \frac{dφ̃_{t}^{∗}Ω}{dt}|_{t=0} = ℒξ̃Ω,

where ℒξ̃Ω ∈ Ω^{1}(P, 𝔤) denotes the Lie derivative of Ω with respect to ξ̃; in particular, φ̃_{t}^{∗}A is a C^{∞} one-parameter family of C^{∞} connection one-forms on P.

Lemma 3.4. Let P be a G bundle with connection A ∈ Ω^{1}(P, 𝔤) over a manifold X. Given a vector field ξ ∈ C^{∞}(T X), let ξ̃ ∈ C^{∞}(T P) be its horizontal lift. If F_{A} ∈ Ω^{2}(P, 𝔤) is the curvature of A, then ℒξ̃A = ιξ̃F_{A}.

Proof. Since ξ̃ is horizontal, then A(ξ̃) = 0 and so for any vector field η ∈ T P, we have (ℒξ̃A)(η) = (ιξ̃dA + dιξ̃A)(η) = dA(η, ξ̃). But F_{A}(η, ξ̃) = dA(η, ξ̃) + \frac{1}{2}[A(η), A(ξ̃)] and so the result follows. □

We also need to consider Lie derivatives of ad P -valued one-forms. Recall that if π : P → X is the bundle projection, there is an injective map π^{∗} : Ω^{1}(X, ad P) ↪ Ω^{1}(P, 𝔤). The one-forms Ω in the image of π^{∗} are characterised by the properties (a) R_{u}^{∗}Ω = Ad (u^{−1})Ω, for all u ∈ G, and (b) Ω(η) = 0 if η ∈ T P is vertical. Hence, the action of φ̃_{t} on Ω^{1}(P, 𝔤) induces an action on Ω^{1}(X, ad P) = Γ(T^{∗}X ⊗ ad P). Thus, if ω ∈ Ω^{1}(X, ad P), we obtain a C^{∞} one-parameter family of C^{∞} ad P -valued one-forms φ̃_{t}^{∗}ω on X with

(3.3)  \frac{dφ̃_{t}^{∗}ω}{dt} |_{t=0} = ℒξ̃ω,

where ℒξ̃ω ∈ Ω^{1}(X, ad P) denotes the Lie derivative of ω with respect to ξ̃.

For the purposes of calculation, it is useful to phrase the preceding discussion in terms of local one-forms on X. It is convenient to choose a system of local sections σ_{α} : U_{α} → P which are parallel with respect to the connection A and vector field ξ, in the sense that A(σ_{α∗}ξ) = 0. For example, one can try to construct σ_{α} by first choosing a section σ_{α}|_{V_{α}}, where V_{α} is a submanifold of U_{α} transverse to the vector field ξ, and then extend by parallel translation along integral curves of ξ to construct a section σ_{α} over a tubular neighbourhood U_{α} of V_{α}. Local sections of this type are described in [U82a, pp. 14-15] and [F-U, pp. 146-147].

Given a system of (A, ξ)-parallel local sections σ_{α}, we have ξ̃ = σ_{α∗}ξ and φ_{t} = σ_{α}^{∗}φ̃_{t} over U_{α}. Hence, for ω ∈ Ω^{1}(X, ad P) we see that σ_{α}^{∗}φ̃_{t}^{∗}ω = φ_{t}^{∗}σ_{α}^{∗}ω and σ_{α}^{∗}ℒξ̃ω = ℒ_{ξ}σ_{α}^{∗}ω on U_{α}, and similarly for A ∈ Ω^{1}(P, 𝔤). Indeed, one can see that the transition functions {u_{αβ}} are constant along the vector field ξ. For if σ_{β} = σ_{α}u_{αβ}, then σ_{β∗}ξ = σ_{α∗}ξ·u_{αβ} +σ_{α}·u_{αβ∗}ξ, which gives A(σ_{β∗}ξ) = Ad (u_{αβ}^{−1})A(σ_{α∗}ξ) + A(σ_{α} · u_{αβ∗}ξ), and thus du_{αβ}(ξ) = 0, since A(σ_{α∗}ξ) = A(σ_{β∗}ξ) = 0 and A(σ_{α} · u_{αβ∗}ξ) = u_{αβ∗}ξ. Here, σ_{α} · u_{αβ∗}ξ is the vector field on P |_{U_{α}} obtained by differentiating the maps G → P given by u ↦ σ_{α}(x)u. When computing Lie derivatives of local connection one-forms or ad P -valued one-forms with respect to a vector field ξ, we shall always require that the local sections σ_{α} be (A, ξ)-parallel.

It is often useful to express ℒξ̃ω in terms of covariant derivatives. Suppose X has a C^{∞} metric g. We have ℒ_{ξ}ω = ι_{ξ}dω +dι_{ξ}ω, or in local coordinates, (ℒ_{ξ}ω)_{μ} = ξ^{ν}∂ω_{μ}/∂x^{ν} + ω_{ν}∂ξ^{ν}/∂x^{μ}. We find that

(3.4)  ℒξ̃ω = ∇_{ξ}^{A,g}ω + ω(∇^{g}ξ),

using normal geodesic coordinates {x^{μ}} and (A, ξ)-parallel local sections {σ_{α}}. In the sequel, we omit the “tildes” to indicate lifts of vector fields or diffeomorphisms on the base to the total space of a principal bundle – this being understood from the context. Remark that if Φ : X → X is a diffeomorphism and ω ∈ Ω^{1}(X, ad P), then we have ℒ_{ξ}Φ^{∗}ω = Φ^{∗}ℒ_{Φ_{∗}ξ}ω.

Let A be a C^{∞} connection on a G bundle P over 𝕊^{4} and let ω ∈ Ω^{1}(𝕊^{4}, ad P). For any t ∈ (−∞, ∞), let δ_{t} be the dilation of ℝ^{4} given by x ↦ e^{t}x and for any p ∈ ℝ^{4}, let τ_{p} be the translation of ℝ^{4} defined by τ_{p} : x ↦ x − p. If δ_{t} and τ_{p} again denote the conformal diffeomorphisms of 𝕊^{4} induced by the chart x = ϕ_{n}^{−1}, then the group C = SO(4)×D×T of rotations, dilations, and translations of ℝ^{4} is identified with the subgroup in Conf(𝕊^{4}, g_{1}) of diffeomorphisms which fix the south pole s ∈ 𝕊^{4}. Setting φ_{t} = δ_{t} or τ_{tp}, these diffeomorphisms are generated by the vector fields

(3.5)  r ≡ x^{μ} \frac{∂}{∂x^{μ}} and − p ≡ −p^{μ} \frac{∂}{∂x^{μ}} .

We always choose p ∈ ℝ^{4} with |p| ≤ 1. We next describe the construction of (A, ξ)-parallel local sections σ_{α} for ξ = r or p.

Considering the group of dilations D, let σ_{n}, σ_{s} be the local sections formed by choosing points in the fibres P |_{n}, P |_{s} and then parallel translating along radial directions from the poles. The transition function u will be constant along the radial directions, du(r) = 0, and the local connection one-forms σ_{α}^{∗}A are in radial gauge. On the other hand, considering the group of translations T, suppose first that p = ∂/∂x^{4} and let σ_{n}|_{𝕊^{3}}, σ_{s}|_{𝕊^{3}} be the local sections formed by parallel translation from the north and south poles of the three-sphere 𝕊^{3} ⊂ 𝕊^{4} defined by the image of the x^{1}x^{2}x^{3}-plane under the map ϕ_{n} : ℝ^{4} → 𝕊^{4} \ {s}. We obtain local sections σ_{n}, σ_{s} by parallel translation along the x^{4}-axis. The transition function u will now be constant along the x^{4}-axis, du(p) = 0, and the local connection one-forms σ_{α}^{∗}A are in a transverse gauge. By a linear change of coordinates, the same argument applies to arbitrary translations.

For the dilations, we have

(3.6)  \frac{dδ_{t}^{∗}ω}{dt} |_{t=0} = ℒ_{r}ω = ι_{r}dω + ω,

using ℒ_{ξ}ω = ι_{ξ}dω + dι_{ξ}ω, or in local coordinates, (ℒ_{ξ}ω)_{μ} = ξ^{ν}∂ω_{μ}/∂x^{ν} + ω_{ν}∂ξ^{ν}/∂x^{μ}. Similarly, for the translations we have

(3.7)  \frac{dτ_{tp}^{∗}ω}{dt} |_{t=0} = −ℒ_{p}ω = −ι_{p}dω,

where p = p^{μ}∂/∂x^{μ}.

For any λ ∈ (0, ∞), let c_{λ} be the diffeomorphism of 𝕊^{4} defined by the chart x = ϕ_{n}^{−1} and the dilation c_{λ} of ℝ^{4} given by x ↦ x/λ. Then c_{λ} = δ_{t} with t = − log λ, and so from Eq. (3.6) we have \frac{∂}{∂λ}c_{λ}^{∗}ω = −\frac{1}{λ}c_{λ}^{∗}ℒ_{r}ω. Similarly, for the translations τ_{q}, q ∈ ℝ^{4}, we see that Eq. (3.7) gives \frac{∂}{∂p}τ_{q}^{∗}ω = −τ_{q}^{∗}ℒ_{p}ω, where ∂/∂p ≡ p^{μ}∂/∂q^{μ} on the left-hand side and using τ_{q+tp} = φ_{tp} ∘ τ_{q} on the right. Combining these actions, we find that

(3.8)  \frac{∂}{∂λ}τ_{q}^{∗}c_{λ}^{∗}ω = −\frac{1}{λ}τ_{q}^{∗}c_{λ}^{∗}ℒ_{r}ω and \frac{∂}{∂p}τ_{q}^{∗}c_{λ}^{∗}ω = −\frac{1}{λ}τ_{q}^{∗}c_{λ}^{∗}ℒ_{p}ω,

Similarly, considering the action of the dilations c_{λ} and translations τ_{q} on connection one-forms, we have

(3.9)  \frac{∂}{∂λ}τ_{q}^{∗}c_{λ}^{∗}A = −\frac{1}{λ}τ_{q}^{∗}c_{λ}^{∗}ι_{r}F_{A} and \frac{∂}{∂p}τ_{q}^{∗}c_{λ}^{∗}A = −\frac{1}{λ}τ_{q}^{∗}c_{λ}^{∗}ι_{p}F_{A}.

These derivative formulas play a significant role in the sequel.

It is convenient at this point to recall Taubes’ definition of a centred connection over the four-sphere [T88, p. 343]. Let A be an g_{1}-ASD connection on a G bundle P with c_{2}(P) = k over 𝕊^{4} with its standard metric g_{1}. Pulling back via the chart x = ϕ_{n}^{−1} : 𝕊^{4} \ {s} → ℝ^{4}, we obtain a δ-ASD connection A on a G bundle P over ℝ^{4} with its standard metric δ. Let Θ denote the flat connection on the product bundle. Suppose A ≠ Θ: then the mass centre q and scale λ are defined by

(3.10)  q = Centre [A] ≡ \frac{1}{8π^{2}k} ∫_{ℝ^{4}} |F_{A}|^{2} d^{4}x,
λ^{2} = Scale^{2}[A] ≡ \frac{1}{8π^{2}k} ∫_{ℝ^{4}} |x − q|^{2}|F_{A}|^{2} d^{4}x.

If A = Θ, we set Centre [A] = 0 and Scale[A] = 0. The connection A is called centred if Centre [A] = 0 and Scale[A] = 1. Eq. (3.10) leads to the following Tchebychev ineqality:

(3.11)  ∫_{|x−q|≥Rλ} |F_{A}|^{2} d^{4}x ≤ 8π^{2}kR^{−2},  R ≥ 1.

Hence, the ball B(q, Rλ) contains A-energy greater than or equal to 8π^{2}k(1 − R^{−2}).

Setting f_{λ,q} = c_{λ} ∘ τ_{q}, we see that Centre [(f_{λ,q}^{−1})^{∗}A] = 0 and Scale[(f_{λ,q}^{−1})^{∗}A] = 1. Let M_{k} denote the moduli space of g_{1}-ASD connections on the bundle P over 𝕊^{4} and let M_{k}^{0} denote the moduli space of centred g_{1}-ASD connections. Note that M_{1}^{0} consists of a single point representing the standard one-instanton over 𝕊^{4}. More generally, the relationship between M_{k} and M_{k}^{0} is explained below:

Proposition 3.5. For any k > 0, the space M_{k}^{0} is a smooth submanifold of M_{k}. Moreover, M_{k} is diffeomorphic to M_{k}^{0} × ℝ^{4} × (0, ∞).

Proof. One argues as in [T88, pp. 343-344] and [T84b, pp. 365-367]. Given [A] ∈ M_{k} with Centre [A] = q and Scale[A] = λ, set f_{λ,q} = c_{λ} ∘ τ_{q}. The map [A] → ([(f_{λ,q}^{−1})^{∗}A], q, λ) then gives the required diffeomorphism. □

3.3. Gluing construction of approximately anti-self-dual connections. We describe the approximate gluing constructions of Donaldson [D86], [D-K], and Taubes [T82], [T84a], [T88], adapted to the case of “bubble trees”. For clarity, we first discuss the construction of approximately anti-self-dual connections over single connected sums. Let X_{0} be our closed, smooth four-manifold with metric g_{0} and injectivity radius ϱ_{0}, and let X_{1} = 𝕊^{4} with its standard round metric g_{1} of radius 1. Let x_{1} be a point in X_{0} and let x_{1n}, x_{1s} denote the north and south poles of X_{1}. Let P_{i} → X_{i} be principal G bundles with c_{2}(P_{i}) = k_{i}, i = 0, 1. Let F X_{0} be the principle SO(4) bundle of oriented, orthnormal frames over X_{0}.

A choice of frame v_{1} ∈ F X_{0}|_{x_{1}} defines a geodesic normal coordinate system ϕ_{1}^{−1} = exp_{v_{1}}^{−1} : B_{1}(ϱ_{0}) → ℝ^{4}. Denote ϕ_{1α} = ϕ_{α}, α = s, n, where ϕ_{α}^{−1} : U_{α} = 𝕊^{4} \ {α} → ℝ^{4} are the standard coordinate charts on the four-sphere. Let B_{1}(r) = B(x_{1}, r) be the open geodesic ball in X_{0} with centre x_{1} and radius r, and let B_{1s}(r) = ϕ_{1s}({x ∈ ℝ^{4} : |x| < r}), an open ball in X_{1} with centre x_{1s}. Let Ω_{1}(r, R) = Ω(x_{1}, r, R) be the open annulus B_{1}(R) \ \overline{B}_{1}(r) centred at x_{1} ∈ X_{0}, with inner radius r and outer radius R; similarly, let Ω_{1s}(r, R) = Ω(x_{1s}, r, R) be the open annulus B_{1s}(R) \ \overline{B}_{1s}(r) in X_{1}.

Let N > 4 be a large parameter, to be fixed later, and let λ_{1} > 0 be a small scale parameter such that λ_{1}^{1/2}N ≪ 1. We define open sets X′_{0} = X_{0} \ \overline{B}_{1}(N^{−1}λ_{1}^{1/2}), X′′_{0} = X_{0} \ \overline{B}_{1}(\frac{1}{2}λ_{1}^{1/2}), and X′′′_{0} = X_{0} \ \overline{B}_{1}(2Nλ_{1}^{1/2}) — the complements in X_{0} of small balls around the point x_{1}. Likewise, define open sets X′_{1}, X′′_{1}, and X′′′_{1} in the sphere X_{1}. Let Ω_{1} denote the annulus Ω_{1}(N^{−1}λ_{1}^{1/2}, Nλ_{1}^{1/2}) in X_{0} and let Ω_{1s} = Ω_{1s}(N^{−1}λ_{1}^{1/2}, Nλ_{1}^{1/2}) be the corresponding annulus in X_{1}. Let c_{1} be the dilation map on ℝ^{4} defined by x ↦ x/λ_{1}. Define balls B′_{1} = B_{1}(Nλ_{1}^{1/2}) and B′′_{1} = B_{1}(2λ_{1}^{1/2}) centred at x_{1} in X_{0} and a diffeomorphism

(3.12)  f_{1} = ϕ_{1n} ∘ c_{1} ∘ ϕ_{1}^{−1} : B′_{1} ⟶ X′_{1}.

Hence, f_{1} identifies the small balls B′_{1} and B′′_{1} in X_{0} with the open sets X′_{1} and X′′_{1} in X_{1}, and restricts to a diffeomorphism f_{1} : Ω_{1} → Ω_{1s}.

We let X be the connected sum X_{0}#_{f_{1}}X_{1}. In §3.5 we define a smooth metric g on X which closely approximates the metrics g_{i} on each summand X′_{i} and such that the map f_{1} : B′_{1} → X′_{1} is conformal. Thus, (X, g) is conformally equivalent to (X_{0}, g_{0}).

Let A_{i} be g_{i}-anti-self-dual connections on the bundles P_{i} → X_{i}, i = 0, 1. The connections A_{0}, A_{1}, together with a choice of points in the fibres P_{0}|_{x_{1}}, P_{1}|_{x_{1s}}, define local sections σ_{1} : B_{1}(ϱ_{0}) → P_{0} and σ_{1s} : X_{1} \ {x_{1n}} → P_{1} by parallel transport along radial geodesics through x_{1}, x_{1s}. Hence, we obtain local trivialisations P_{0}|_{B_{1}} ≃ B_{1} × G and P_{1}|_{B_{1s}} ≃ B_{1s} × G.

Let b_{1} ≥ 4Nλ_{1}^{1/2} be a small parameter, b_{1} < \frac{1}{4} min\{1, ϱ_{0}\}: we will eventually set b_{1} = 4Nλ_{1}^{1/2}. Choose cutoff functions ψ_{i} on X_{i} such that 0 ≤ ψ_{i} ≤ 1, with ψ_{0} = 1 on X_{0} \ B_{1}(b_{1}), ψ_{0} = 0 on B_{1}(b_{1}/2), and similarly for ψ_{1} on X_{1}. We let A′_{0} = ψ_{0}A_{0} be the C^{∞} connection on the bundle π_{0} : P_{0} → X_{0} defined by

(3.13)  A′_{0} = \{ \begin{matrix}A_{0} & on P_{0}|_{X_{0}\B_{1}(b_{1})} \\ π_{0}^{∗}(ψ_{0}σ_{1}^{∗}A_{0}) & on P_{1}|_{B_{1}(b_{1})}. \end{matrix}

Of course, we have the analogous definition for the C^{∞} connection A′_{1} over X_{1} and we obtain almost anti-self-dual connections which are flat on the balls B′_{1}, B′_{1s}.

To construct the cutoff functions ψ_{i}, choose a C^{∞} bump function ζ on ℝ^{1} such that ζ(t) = 1 for t ≥ 1 and ζ(t) = 0 for t ≤ 1/2. Define a C^{∞} cutoff function ψ_{b} on ℝ^{4} by ψ_{b}(x) = ζ(|x|/b), for any b > 0. Set ψ_{0} = (ϕ_{1}^{−1})^{∗}ψ_{b_{1}} and extend by 1 on X_{0} \ B(x_{1}, b_{1}) and by zero on B(x_{1}, b_{1}/2) to give ψ_{0} ∈ C^{∞}(X_{0}); likewise, set ψ_{1} = (ϕ_{1s}^{−1})^{∗}ψ_{b_{1}} and extend to give ψ_{1} ∈ C^{∞}(X_{1}). Each ψ_{i} extends by zero to give a C^{∞} cutoff function on the connected sum X.

Choose a G-isomorphism ρ_{1} ∈ Gl_{x_{1}}, where Gl_{x_{1}} ≡ Hom_{G}(P_{0}|_{x_{1}}, P_{1}|_{x_{1s}}) ≃ G is the space of “gluing parameters”. Using the connections A_{i} over the small \frac{1}{2}b_{1}-balls, spread out the fibre isomorphism ρ_{1} to give a bundle isomorphism ρ̃_{1} : P_{0}|_{Ω_{1}} → P_{1}|_{Ω_{1s}} covering the diffeomorphism f_{1} : Ω_{0} → Ω_{1}. Thus, σ_{1}ρ̃_{1} = f_{1}^{∗}σ_{1s} on Ω_{1}. We define the smooth connected-sum bundle P → X with second Chern class c_{2}(P) = k = k_{0} + k_{1} by setting P |_{X′_{0}} = P_{0}|_{X′_{0}} and P |_{X′_{1}} = P_{1}|_{X′_{1}}. Note that the bundle P is defined by transition functions independent of the scale λ_{1}. We define a smooth connection A′ = A′_{0}#A′_{1} on P → X by setting A′ = A′_{i} on each summand X′_{i}.

If Γ_{A_{i}} are the isotropy groups of the connections A_{i} and Γ = Γ_{A_{0}} × Γ_{A_{1}}, then we recall that the gluing construction gives a bijection between the gauge equivalence classes [A′(ρ_{1})] in B_{X,k} and Gl_{x_{1}}/Γ [D-K, p. 286].

Using the diffeomorphism f_{1} : B′_{1} → X′_{1}, we pull back the bundle P over X to a bundle P̂ over X_{0}, given by P̂|_{X′_{0}} = P_{0}|_{X′_{0}} and P̂|_{B′_{1}} = f_{1}^{∗}P_{1}|_{B′_{1}}. We have an induced system of local sections of P̂|_{B′_{0}} given near x_{1} by σ̂_{1n} = f_{1}^{∗}σ_{1n} : B′_{1} → P, σ̂_{1s} = f_{1}^{∗}σ_{1s} : B′_{1} \ {x_{1}} → P, and σ̂_{1} = σ_{1} : Ω_{1}(N^{−1}λ_{1}^{1/2}, ϱ_{0}) → P. The corresponding transition functions û_{1} = f_{1}^{∗}u_{1} : B′_{1} \ {x_{0}} → G and ρ̃_{1} : Ω_{1} → G are determined by σ̂_{1s} = σ̂_{1n}û_{1} on B′_{1} \ {x_{0}} and σ_{1}ρ̃_{1} = f_{1}^{∗}σ̂_{1s} on Ω_{1}.

On the pull-back bundle P̂ → X_{0} we define the corresponding smooth pull-back connection Â′ by setting Â′ = A′_{0} on P̂|_{X′_{0}} and Â′ = f_{1}^{∗}A′_{1} on P̂|_{B′_{1}}. We obtain local connection 1-forms for Â′ over X_{0} given by σ̂_{1n}^{∗}Â′ = f_{1}^{∗}σ_{1n}^{∗}A′_{1} on the ball B′_{1}, σ̂_{1}^{∗}Â′ = σ_{1}^{∗}A′_{0} on the annulus Ω_{1}(N^{−1}λ_{1}^{1/2}, ϱ_{0}), and σ̂_{1s}^{∗}A′ = f_{1}^{∗}σ_{1s}^{∗}A′_{1} on the punctured ball B′_{1} \ {x_{1}}.

On the annulus Ω_{1} we have σ̂_{1s}^{∗}Â′ = σ_{1}^{∗}Â′ = 0, and since

(3.14)  σ̂_{1s}^{∗}Â′ = ρ̃_{1}^{−1}σ̂_{1}^{∗}Â′_{0}ρ̃_{1} + ρ̃_{1}^{−1}dρ̃_{1} on Ω_{1},

we see that dρ̃_{1} = 0 on Ω_{1} and so ρ̃_{1} is constant on Ω_{1}. The transition function û_{1} on B′_{1} \ {x_{0}} is independent of λ_{1}, since u_{1} on X_{1} \ {x_{1n}, x_{1s}} is constant along geodesics connecting the north and south poles. Thus, the bundle P̂ is defined by transition functions which are constant with respect to λ_{1}.

We now generalise the preceding discussion to give a construction of approximately anti-self-dual connections over multiple connected sums. The description we give here is closely related to Taubes’ iterated gluing construction [T88, §4]. The construction parallels the description of the ends of the bubble tree compactification M_{X_{0},k}(g_{0}) described in Chapter 4.

It is convenient at this point to introduce some terminology. Let I = (i_{1}, …, i_{r}) denote a multi-index of positive integers. The length of I is r; we regard 0 as a multi-index of length zero. Given I = (i_{1}, …, i_{r}), we let I_{−} = (i_{1}, …, i_{r−1}); we will often denote a multi-index of the form (i_{1}, …, i_{r+1}) by I_{+} or if we wish to be more specific, by Ij, where j = i_{r+1} > 0 or s, n (indicating north or south poles of 𝕊^{4}), with a slight abuse of notation. Let ℐ be an oriented tree with a finite set of vertices {I}, including a base vertex 0, and a set of edges {(I, I_{+})}. If I = (i_{1}, …, i_{r}) and I = (j_{1}, …, j_{t}), then we say I < J if r < t and J = (i_{1}, …, i_{r}, j_{r+1}, …, j_{t}). The valence of each vertex I is the number of edges emanating from that vertex. The height of the tree ℐ is the number of levels – the length of the longest multi-index minus one. With respect to a given vertex I, the edge (I_{−}, I) is called incoming and the edge (I, I_{+}) is called outgoing.

The construction of a C^{∞}, approximately g-anti-self-dual connection A′ of second Chern class k ≥ 1, associated with a tree ℐ, requires the following data:

Data 3.6. Gluing data for approximately anti-self-dual connections.

(1) To each vertex I, we associate a g_{I}-anti-self-dual connection A_{I} on a G bundle P_{I} → X_{I} with c_{2}(P_{I}) = k_{I} ≥ 0. If I = 0, then X_{0} is the base four-manifold with metric g_{0}, while if I > 0, then X_{I} = 𝕊^{4} with its standard round metric g_{I} ≡ g_{1} of radius 1.

(2) To each edge (I_{−}, I), we associate the data (b_{I}, λ_{I}, ρ_{I}, x_{I}, v_{I}) given by the (i) Connection cutoff parameter b_{I}; (ii) Scale parameter λ_{I}; (iii) Bundle gluing parameter ρ_{I} ∈ Gl_{x_{I}}, where Gl_{x_{I}} = Hom(P_{I_{−}}|_{x_{I}} , P_{I}|_{x_{Is}}); (iv) Centre or gluing site x_{I} ∈ X_{I_{−}};

(v) Frame v_{I} ∈ F X_{0}|_{x_{I}} if I_{−} = 0.

(3) Constants b_{0}, d_{0}, λ_{0}, N.

For convenience, if I_{+} = Is, we denote b_{Is} = b_{I}, λ_{Is} = λ_{I}, N_{Is} = N, and ρ_{Is} = ρ_{I}. We let x_{In}, x_{Is} denote the north and south poles of the spheres X_{I} = 𝕊^{4}. If I_{−} > 0, then x_{I} ≡ ϕ_{I_{−}n}(q_{I}) ∈ X_{I}, where q_{I} ∈ ℝ^{4}. Define

(3.15)  \overline{b} = max_{I∈ℐ}b_{I} and \overline{λ} = max_{I∈ℐ}λ_{I}.

The gluing data should satisfy the following constraints:

Condition 3.7. Gluing data constraints.

(1) Scales: 4Nλ_{I}^{1/2} ≤ b_{I} < \frac{1}{4} min\{1, ϱ_{0}, d_{0}\}, 4 < N_{0} ≤ N, and 0 < λ_{I} ≤ λ_{0};

(2) Separation of centres: Suppose x_{I}, x_{I′} ∈ X_{I_{−}}.

(i) If I_{−} = 0, then dist_{g_{0}}(x_{I}, x_{I′}) > 4(b_{I} + b_{I′}), (ii) If I_{−} > 0, then |q_{I} − q_{I′}| > 4(b_{I} + b_{I′});

(3) Topology: ∑_{I∈ℐ} k_{I} = k and k_{I} > 0 for some I > 0.

Remark 3.8. Definition 3.6, together with the constraints of Condition 3.7 should be compared with the definition of “bubble tree ideal” connections in §4.3. The requirements on the scales and separation of centres are in place simply to ensure that the different gluing regions do not interfere with one another.

The gluing procedure now generalises to give a C^{∞} family of approximately g-anti-self-dual connections A′ = #_{I∈ℐ}A′_{I} on a bundle P over a multiple connected sum X = #_{I∈ℐ}X′_{I}. First, consider the definition of coordinate charts, open balls, and annuli in X_{0}. If I_{−} = 0, let ϕ_{I}^{−1} = exp_{v_{I}}^{−1} : B(x_{I}, ϱ_{0}) → ℝ^{4} be a geodesic normal coordinate chart defined by a point v_{I} in the oriented frame bundle fibre F X_{0}|_{x_{I}}. Let B_{I}(r) = B(x_{I}, r) be the open geodesic ball in X_{0} with centre x_{I} and radius r.

Turning to the four-spheres X_{I}, for any I > 0, let ϕ_{Iα} = ϕ_{α}, α = s, n be the standard inverse coordinate charts on X_{I}. Define open neighbourhoods in X_{I} by

(3.16)  B_{Is}(r) = B(x_{Is}, r) = ϕ_{Is} ({x ∈ ℝ^{4} : |x| < r}) and
B_{I_{+}}(r) = B(x_{I_{+}}, r) = ϕ_{In} ({x ∈ ℝ^{4} : |x − q_{I_{+}}| < r}) .

Let Ω_{I}(r, R) = Ω(x_{I}, r, R) be the open annulus B_{I}(R) \ \overline{B}_{I}(r) centred at x_{I} ∈ X_{I_{−}}, with inner radius r and outer radius R.

Define small balls B′_{I} = B(x_{I}, Nλ_{I}^{1/2}) and annuli Ω_{I} = Ω(x_{I}, Nλ_{I}^{1/2}, Nλ_{I}^{1/2}) in X_{I_{−}}, I > 0. The open subset X′_{I_{−}} is the complement in X_{I_{−}} of the balls \overline{B}_{I}(N^{−1}λ_{I}^{1/2}), the open

subset X′′_{I_{−}} is the complement in X_{I_{−}} of the balls \overline{B}_{I}(\frac{1}{2}λ_{I}^{1/2}), and the open subset X′′′_{I_{−}} is

the complement in X_{I_{−}} of the balls \overline{B}_{I}(2Nλ_{I}^{1/2}) We define identification maps f_{I} by

(3.17)  f_{I} = ϕ_{In} ∘ c_{I} ∘ ϕ_{I}^{−1} : B′_{I} ⟶ X′_{I},

where c_{I} is the dilation x → x/λ_{I} on ℝ^{4}. The maps ϕ_{I} above are local coordinate charts on X_{I_{−}} given by

(3.18)  ϕ_{I}^{−1} = \{ \begin{matrix}exp_{v_{I}}^{−1} & if I_{−} = 0 \\ τ_{I} ∘ ϕ_{I_{−}n}^{−1} & if I_{−} > 0, \end{matrix}

where τ_{I} is the translation x → x − q_{I} on ℝ^{4}. The charts ϕ_{I}^{−1} = exp_{v_{I}}^{−1} may be replaced by ϕ̄_{I}^{−1} = τ_{p_{I}} ∘ exp_{v_{I}}^{−1}, |p_{I}| ≪ ϱ_{0}, if we wish to compute derivatives with respect to the centres x_{I} in X_{0}. For notational consistency, we let f_{0} denote the identity map on X_{0}.

Using the diffeomorphisms f_{I} : Ω_{I} → Ω_{Is} we obtain a connected sum X = #_{I∈ℐ}X′_{I}. We again defer to §3.5 for the precise definition of a metric g on X closely approximating the metrics g_{I} on the summands X′_{I} and such that the maps f_{I} : B′_{I} → X′_{I} are conformal. With this choice of metric, the connected sum (X, g) is conformally equivalent to (X_{0}, g_{0}).

We have a local section σ_{I} of P_{I_{−}} defined by a choice of point in the fibre P_{I_{−}}|_{x_{I}} and A_{I_{−}}-parallel translation from x_{I}; similarly, we have local sections σ_{In}, σ_{Is} of P_{I} defined by a choice of points in the fibres P_{I}|_{x_{In}}, P_{I}|_{x_{Is}} and A_{I}-parallel translation from x_{In}, x_{Is}. These sections provide local trivialisations P_{I_{−}}|_{B_{I}(ϱ_{0})} ≃ B_{I}(ϱ_{0}) × G and P_{I}|_{X_{I}\{x_{In}}} ≃ X_{I} \ {x_{In}} × G. Define C^{∞} cutoff functions ψ_{I} on each summand X_{I} by setting

(3.19)  ψ_{I} ≡ (ϕ_{Is}^{−1})^{∗}ψ_{b_{I}} ∏_{I_{+}}(ϕ_{I_{+}}^{−1})^{∗}ψ_{I_{+}} on X_{I},

where the factor (ϕ_{Is}^{−1})^{∗}ψ_{b_{I}} is omitted when I = 0. Note that ψ_{I} = 0 on the balls B_{Is}(b_{I}/2) and B_{I_{+}}(b_{I_{+}}/2) in X_{I} and smoothly extends by 1 on the complement of the balls B_{Is}(b_{I}) and B_{I_{+}}(b_{I_{+}}) in X_{I}. Lastly, extend each ψ_{I} by zero to give a C^{∞} cutoff function on the connected sum X. Setting A′_{I_{−}} = ψ_{I_{−}}A_{I_{−}}, A′_{I} = ψ_{I}A_{I}, we obtain C^{∞} almost anti-self-dual connections A′_{I_{−}}, A′_{I} which are flat on the balls B_{I}(b_{I}/2), B_{Is}(b_{I}/2).

The gluing parameter ρ_{I} provides an isomorphism of the fibres : P_{I_{−}}|_{x_{I}} ≃ P_{I}|_{x_{Is}}. Using the connections A_{I_{−}}, A_{I}, this identification is extended to give a bundle isomorphism ρ̃_{I} : P_{I_{−}}|_{Ω_{I}} → P_{I}|_{Ω_{Is}} covering f_{I}. Using these identification maps we obtain a connectedsum G bundle P → X with c_{2}(P) = k and whose transition functions are constant with respect to the scales λ_{I}. The cutoff connections A′_{I} on P_{I} patch together to give a C^{∞} connection A′ on P. As before, the connection A′ on the connected-sum bundle P over X pull back via the maps f_{I} to give a connection Â′ on a bundle P̂ over X_{0}.

Lastly, we record some estimates for the connections A′ when restricted to a summand X′_{I}. For this and later purposes, we define the following Sobolev norms: Let ∇^{g_{I}} denote the Levi-Civita connection on T X_{I} defined by the metric g_{I}, so that if f ∈ C^{∞}(X_{I}), then

(3.20)  ∥f∥_{L_{n}^{p}(X_{I},g_{I})} = ∑_{i=0}^{n} ∥(∇^{g_{I}})^{i}f∥_{L^{p}(X_{I},g_{I})},

for any 1 ≤ p ≤ ∞ and integer n ≥ 0. Similarly, if α ∈ Ω^{l}(X_{I}, ad P_{I}), then

(3.21)  ∥α∥_{L_{n}^{p}(X_{I},A_{I},g_{I})} = ∑_{i=0}^{n} ∥(∇^{A_{I},g_{I}})^{i}α∥_{L^{p}(X_{I},g_{I})}.

It is important to note that these norms will depend only on a set of fixed connections, {A_{I}}_{I∈ℐ}, and a set of fixed metrics {g_{I}}_{I∈ℐ}.

Recalling that A′_{I} = ψ_{I}A_{I}, define one-forms a_{I} ∈ Ω^{1}(X_{I}, ad P_{I}) by setting A_{I} = A′_{I} + a_{I}. Thus

a_{I} = \{ \begin{matrix}(1 − ψ_{I})σ_{I_{+}}^{∗}A_{I} & on B_{I_{+}}(b_{I_{+}}), \\ 0 & on X_{I} \ ⋃_{I_{+}} B_{I_{+}}(b_{I_{+}}). \end{matrix}

With the aid of bounds for the derivatives of the cutoff functions ψ_{J} for C = C(g_{J}) and J = I_{−} or I,

(3.22)  |dψ_{J}|_{g_{J}} ≤ Cb_{J}^{−1} on Ω_{J}(b_{J}/2, b_{J}) and ∥dψ_{J}∥_{L^{4}(X_{J} ,g_{J})} ≤ C,

standard arguments then give the following estimates:

Lemma 3.9. Let 1 ≤ p < ∞. Then there exists a constant C = C(A_{I}, g_{I}, p) such that (a) ∥a_{I}∥_{L^{∞}(X_{I},g_{I})} ≤ C\overline{b} and ∥a_{I}∥_{L^{p}(X_{I},g_{I})} ≤ C\overline{b}^{4/p+1},

(b) ∥F (A′_{I})∥_{L^{∞}(X_{I},g_{I})} ≤ C and ∥F^{+,g_{I}} (A′_{I})∥_{L^{p}(X_{I},g_{I})} ≤ C\overline{b}^{4/p}.

3.4. Approximate gluing maps. Adopting a more global perspective, the construction of §3.3 yields a family of “approximate gluing maps”, 𝒥′ : 𝒯/Γ → B_{X,k}^{∗} and 𝒥̂′ : 𝒯/Γ → B_{X_{0},k}^{∗}, which we describe in this section. We first recall that the standard Kuranishi models give the required parametrisations for neighbourhoods of points [A_{I}] in M_{X_{I},k_{I}} (g_{I}). Let A_{I} be a g_{I}-anti-self-dual connection over X_{I}, with isotropy group Γ_{A_{I}} and H_{A_{I}}^{2} = 0. For a small enough open neighbourhood T_{A_{I}} of 0 ∈ H_{A_{I}}^{1} , we have smooth Γ_{A_{I}} -equivariant maps

α_{I} : T_{A_{I}} ⟶ ker d_{A_{I}}^{∗,g_{I}} ⊂ Ω^{1}(X_{I}, ad P_{I}) (3.23)

solving the g_{I}-anti-self-dual equation F^{+,g_{I}} (A_{I} + α_{I}(t_{I})) = 0, t_{I} ∈ T_{A_{I}}. Setting A_{I}(t_{I}) = A_{I} + α_{I}(t_{I}), we obtain a homeomorphism

(3.24)  ϑ_{I} : T_{A_{I}} /Γ_{A_{I}} ⟶ U_{A_{I}} ,  t_{I} ⟼ [A_{I}(t_{I})],

onto an open neighbourhood U_{A_{I}} of [A_{I}] ∈ M_{X_{I},k_{I}}. If A_{I} is the product connection, Θ, then Γ_{A_{I}} = SU(2) and so H_{A_{I}}^{0} ≠ 0, while H_{A_{I}}^{1} = 0. If A_{I} is a non-trivial reducible connection, then Γ_{A_{I}} = 𝕊^{1} and H_{A_{I}}^{0} ≠ 0: we have a homeomorphism ϑ_{I} : T_{A_{I}} /Γ_{A_{I}} → U_{A_{I}} and a diffeomorphism ϑ_{I} : (T_{A_{I}} \ {0})/Γ_{A_{I}} → U_{A_{I}} \ [A_{I}]. Finally, if A_{I} is irreducible, then Γ_{A_{I}} = (±1) and H_{A_{I}}^{0} = 0: in this case we have a diffeomorphism ϑ_{I} : T_{A_{I}} /Γ_{A_{I}} → U_{A_{I}}.

We now dispose of the construction of neighbourhoods of reducible connections in M_{X_{0},k}(g_{0}). Recall that the reducible connections in M_{X_{0},k}(g_{0}) are in one-to-one correspondence with pairs {±c}, where c ∈ H^{2}(X_{0}, ℤ) satisfies c^{2} = k. In particular, there are only finitely many and so to describe a neighbourhood of any such reducible connection [A] ∈ M_{X_{0},k}(g_{0}), we may employ the Kuranishi model ϑ_{A} : T_{A}/Γ_{A} → U_{A}.

We now describe the approximate gluing maps 𝒥′ and 𝒥̂′, beginning with the parameter spaces 𝒯/Γ. First, with the centres {x_{I}} and scales {λ_{I}} held fixed, the parameter spaces T_{A_{I}} and Gl_{x_{I}} combine to give a C^{∞} manifold

(3.25)  T ≡ T_{A_{0}} × ∏_{I∈ℐ}(T_{A_{I}} × Gl_{x_{I}}) ,

parametrising a “small” family of approximately anti-self-dual connections. Then

(3.26)  Γ ≡ Γ_{A_{0}} × ∏_{I∈ℐ}Γ_{A_{I}}

acts freely on T and T/Γ is a C^{∞} manifold. If we allow the centres, now denoted y_{I}, to move over disjoint balls B(x_{I}, r_{0}) ⊂ X_{I_{−}} and allow the scales λ_{I} to vary in the interval (0, λ_{0}), the parameter space of Eq. (3.25) is augmented to give a C^{∞} manifold

(3.27)  𝒯 ≡ T_{A_{0}} × ∏_{I∈ℐ}(T_{A_{I}} × Gl_{x_{I}} × B(x_{I}, r_{0}) × (0, λ_{0})) ,

parametrising a “large” family of approximately g-anti-self-dual connections. Again, Γ acts freely on 𝒯 and 𝒯/Γ is a C^{∞} manifold. We fix local trivialisations of the frame bundle F X_{0} over the balls B(x_{I}, r_{0}) and these provide smooth families of geodesic normal coordinate charts on X_{0}.

We note that the almost anti-self-dual connections A′ produced by §3.3 are indeed irreducible:

Lemma 3.10. Let A′ be a connection on the G bundle P over X defined by Data 3.6 and Condition 3.7. Then A′ is irreducible, that is H_{A′}^{0} = 0, for small enough b_{0} and large enough N_{0}.

The Lemma follows from Aronszajn’s unique continuation principle for solutions to Δ_{A′}η = 0 via standard methods, so the proof is omitted. Hence, the approximate gluing construction of §3.3 gives a C^{∞} map

(3.28)  𝒥′ : 𝒯/Γ ⟶ B_{X,k}^{∗},  t ⟼ [A′(t)],

where B_{X,k}^{∗} has the structure of an L_{n}^{2} Hilbert manifold, n ≥ 3. Moreover, 𝒥′ is a C^{∞} submersion onto its image; see §5.2. We refer to 𝒥′ as an approximate gluing map over X and its image 𝒰′ ⊂ B_{X,k}^{∗} as an approximate gluing neighbourhood.

The dimension of the parameter space 𝒯/Γ is given by

(3.29)  dim 𝒯/Γ = dim H_{A_{0}}^{1} − dim H_{A_{0}}^{0} + ∑_{I>0}(dim H_{A_{I}}^{1} − dim H_{A_{0}}^{0} + 8),

since each factor Gl_{x_{I}} × B(x_{I}, r_{0}) × (0, λ_{0}) has dimension 8, dim H_{A_{I}}^{0} = dim Γ_{A_{I}}, and H_{A_{I}}^{2} = 0 for all I ≥ 0 by hypothesis. Families of centred g_{I}-anti-self-dual connections A_{I} ∈ M_{X_{I},k_{I}}^{0} (g_{I}) are parametrised by small balls T_{A_{I}}^{0} and thus, we obtain a C^{∞} parameter space

(3.30)  𝒯^{0} ≡ T_{A_{0}} × ∏_{I∈ℐ}(T_{A_{I}}^{0} × Gl_{x_{I}} × B(x_{I}, r_{0}) × (0, λ_{0})) ,

with C^{∞} quotient 𝒯^{0}/Γ of dimension equal to dim M_{X,k}(g). The map 𝒥′ : 𝒯^{0}/Γ → B_{X,k}^{∗} is a C^{∞} embedding; see §5.2.

Lastly, using the conformal diffeomorphisms f_{I}, the bundle P over X pulls back to a bundle P̂ over X_{0}. The gluing construction now produces an approximately g_{0}-anti-self-dual connection Â′ in B_{X_{0},k}^{∗}. The map 𝒥′ of Eq. (3.28) pulls back to a C^{∞} map

(3.31)  𝒥̂′ : 𝒯/Γ ⟶ B_{X_{0},k}^{∗},  t ⟼ [Â′(t)].

Again, 𝒥̂′ is a C^{∞} submersion onto its image and and is a C^{∞} embedding when the parameter space 𝒯/Γ is replaced by the smaller parameter space 𝒯^{0}/Γ; see §5.2. As before, the image 𝒱′ of 𝒥̂′ in B_{X_{0},k}^{∗} is called an approximate gluing neighbourhood.

3.5. Metrics on connected sums. In this section we define a conformal structure [g] on the connected sum X = #_{I∈ℐ}X_{I}. This is accomplished by replacing the standard round metric g_{I} on each spherical summand X′_{I} by a quasi-conformally equivalent metric g̃_{I} so that the identification maps f_{I} : B′_{I} → X′_{I} are conformal. We then construct a C^{∞} metric g on X in the conformal class [g] = [g_{0}] and compare the resulting L^{p} norms for the different possible metrics on each summand X′_{I}. Our construction is modelled after the constructions of Donaldson and Taubes for metrics on connected sums – see [D86, p. 322], [D-K, p. 293], [D-S], and [T92]. The metric g depends on the choice of fixed base metric g_{0}, fixed neck width parameter N, scales λ_{I}, centres x_{I}, and frames v_{I}. We also obtain bounds for the derivatives of g with respect to λ_{I} and x_{I}.

With respect to a geodesic normal coordinate system x = ϕ_{i_{1}}^{−1} on B_{i_{1}}(ϱ_{0}) ⊂ X_{0}, the covariant components of g_{0} satisfy

(3.32)  (ϕ_{i_{1}}^{∗}g_{0})_{μν}(0) = δ_{μν} and \frac{∂(ϕ_{i_{1}}^{∗}g_{0})_{μν}}{∂x^{α}} (0) = 0,
|(ϕ_{i_{1}}^{∗}g_{0})_{μν} − δ_{μν}|(x) ≤ c|x|^{2} and |\frac{∂(ϕ_{i_{1}}^{∗}g_{0})_{μν}}{∂x^{α}} | (x) ≤ c|x|,  |x| < ϱ_{0}/2,

for some c = c(g_{0}) [K-N]. The analogous relations hold for the contravariant components of g_{0}. We now define a conformal structure [g] on X:

Definition 3.11. The conformal structure [g] on X is defined by the C^{∞} metric g_{0} on X′_{0} and a choice of C^{∞} metric g̃_{I} on each summand X′_{I}, I > 0, given by

(ϕ_{In}^{∗}g̃_{I})_{μν}(x) ≡ \{ \begin{matrix}h_{1}^{2}(x)(ϕ_{I}^{∗}g_{0})_{μν}(λ_{I}x) & if I_{−} = 0 \\ h_{1}^{2}(x)h_{1}^{−2}(λ_{I}x + q_{I})(ϕ_{I}^{∗}g̃_{I_{−}})_{μν}(λ_{I}x) & if I_{−} > 0, \end{matrix}

where |x| < Nλ_{I}^{−1/2}. For convenience, we let g_{I} ≡ g_{1} denote the standard metric on X_{I} and let g̃_{0} ≡ g_{0} denote the metric on X_{0}.

Definition 3.11 provides the following expression for g̃_{I}:

(3.33)  (ϕ_{In}^{∗}g̃_{I})_{μν}(x) = h_{1}^{2}(x)(ϕ_{i_{1}}^{∗}g_{0})_{μν}(y(x)),  |x| < Nλ_{I}^{−1/2},

where

(3.34)  y(x) = ϕ_{i_{1}}^{−1} ∘ f_{i_{1}}^{−1} ∘ ⋯ ∘ f_{I}^{−1} ∘ ϕ_{In}(x)
= λ_{i_{1}}(λ_{i_{1}i_{2}}(⋯ (λ_{I_{−}}(λ_{I}x + q_{I}) + q_{I_{−}}) ⋯) + q_{i_{1}i_{2}}).

The map f_{I} : B′_{I} → X′_{I} is now conformal with respect to the metrics g̃_{I_{−}} on B′_{I} ⊂ X′_{I_{−}} and g̃_{I} on X′_{I}:

(ϕ_{I}^{∗}f_{I}^{∗}g̃_{I})_{μν}(x) = \{ \begin{matrix}λ_{I}^{−2}h_{1}^{2}(x/λ_{I})(ϕ_{I}^{∗}g̃_{0})_{μν}(x) & if I_{−} = 0 \\ λ_{I}^{−2}h_{1}^{2}(x/λ_{I})h_{1}^{−2}(x + q_{I})(ϕ_{I}^{∗}g̃_{I_{−}})_{μν}(x) & if I_{−} > 0, \end{matrix}

where |x| < Nλ_{I}^{1/2}. Thus, f_{I}^{∗}g̃_{I} is conformally equivalent to the metric g_{I_{−}} on Ω_{I} and so we obtain a conformal structure [g] on X = #_{I∈ℐ}X_{I}.

We must verify that g̃_{I} is a good approximation to the standard round metric g_{I} on X′_{I} for λ_{i_{1}} small.

Lemma 3.12. For any I > 0, the metric g̃_{I} converges to g_{I} in C^{∞} on compact subsets of X_{I} \ {x_{Is}} as λ_{i_{1}} → 0. Moreover, we have the following bounds:

(a) For any integer l ≥ 0, there is a constant c = c(g_{0}, l) such that

|\frac{∂^{l}(ϕ_{In}^{∗}g̃_{I})_{μν}}{∂x^{α_{1}} ⋯ ∂x^{α_{l}}} − \frac{∂^{l}(ϕ_{In}^{∗}g_{I})_{μν}}{∂x^{α_{1}} ⋯ ∂x^{α_{l}}} | ≤ cN^{2}λ_{i_{1}}h_{1}^{2}(x), |x| < Nλ_{I}^{−1/2}.

The analogous bounds hold for the contravariant components (ϕ_{In}^{∗}g̃_{I})^{μν}, provided h_{1}^{2}(x) is replaced by h_{1}^{−2}(x).

(b) Let ∗g̃_{I} denote the Hodge star operator for g̃_{I}. Then there is a constant c = c(g_{0}) such that

∥ ∗g̃_{I} ζ − ∗_{g_{I}} ζ∥_{L^{∞}(X′_{I},g_{I})} ≤ cN^{2}λ_{i_{1}}∥ζ∥_{L^{∞}(X′_{I},g_{I})}, ζ ∈ Ω^{2}(X′_{I}, ad P_{I}).

Proof. (a) This follows easily from Eq. (3.32) and Definition 3.11. (b) This follows immediately from (a) and the definition of the Hodge star operator. □

We will also require bounds for the derivatives of g̃_{J} with respect to the scales λ_{I} and centres x_{I}. The following estimates will suffice for our application.

Lemma 3.13. If 0 < I ≤ J, there is a constant c = c(g_{0}, J) such that the following bounds hold.

(a) For any |x| < Nλ_{J}^{−1/2},

|\frac{∂(ϕ_{Jn}^{∗}g̃_{J})_{μν}}{∂λ_{I}} | (x) ≤ \{ \begin{matrix}cλ_{i_{1}}h_{1}^{2}(x) & if I < J and |I| = 1 \\ cλ_{i_{1}}^{2}h_{1}^{2}(x) & if I < J and |J| ≥ 2 \\ cN^{2}h_{1}^{2}(x) & if I = J and |J| = 1 \\ cλ_{i_{1}}^{2}|x|h_{1}^{2}(x) ≤ cNλ_{i_{1}}^{2}λ_{I}^{−1/2}h_{1}^{2}(x) & if I = J and |J| ≥ 2. \end{matrix}

(b) If ∂/∂p_{I} ≡ p_{I}^{α}∂/∂q_{I}^{α}, then for any |x| < Nλ_{J}^{−1/2},

|\frac{∂(ϕ_{Jn}^{∗}g̃_{J})_{μν}}{∂p_{I}} | (x) ≤ \{ \begin{matrix}cNλ_{i_{1}}^{1/2}h_{1}^{2}(x) & if I = J and |J| = 1 \\ cλ_{i_{1}}^{2}h_{1}^{2}(x) & if I ≤ J and |J| ≥ 2. \end{matrix}

The analogous bounds in (a) and (b) hold for the contravariant components of g̃_{J}, if h_{1}^{2}(x) is replaced by h_{1}^{−2}(x).

(c) For any ζ ∈ Ω^{2}(X′_{J}, ad P_{J}), then

∥\frac{∂∗g̃_{J}}{∂λ_{I}} ζ∥_{L^{∞}(X′_{J} ,g_{J})} ≤ \{ \begin{matrix}cNλ_{I}^{−1/2}∥ζ∥_{L^{∞}(X′_{J} ,g_{J})} & if I = J and |J| ≥ 2 \\ cN^{2}∥ζ∥_{L^{∞}(X′_{J} ,g_{J})} & otherwise, \end{matrix}

∥\frac{∂∗g̃_{J}}{∂p_{I}} ζ∥_{L^{∞}(X′_{J} ,g_{J})} ≤ cNλ_{i_{1}}^{1/2}∥ζ∥_{L^{∞}(X′_{J} ,g_{J})}.

Proof. (a) The inequalities follow from Eq. (3.32) and Definition 3.11.

(b) The proof is similar. When |I| = 1, we recall that the normal geodesic chart ϕ_{i_{1}} ≡ exp_{v_{i_{1}}} is replaced by ϕ̄_{i_{1}} ≡ exp_{v_{i_{1}}} ∘τ_{q_{i_{1}}} in order to compute the required derivative at q_{i_{1}} = 0 (corresponding to x_{i_{1}} = ϕ_{i_{1}}(0)). The estimates follow immediately from (a) and (b). □

We next define an honest C^{∞} metric g on X. Consider a neck Ω_{I} = f_{I}^{−1}(Ω_{Is}) labelled by the multi-index I. We replace the metric g̃_{I_{−}} on the annulus Ω_{I} and replace the metric g̃_{I} on the annulus Ω_{Is} by conformally equivalent metrics m_{I_{−}} g̃_{I_{−}} and m_{I} g̃_{I} so that

(3.35)  m_{I_{−}}g_{I_{−}} = f_{I}^{∗}(m_{I}g_{I}) on Ω_{I}.

Hence, the metrics m_{I_{−}} g̃_{I_{−}} and m_{I} g̃_{I} agree on the neck and patch together to give a C^{∞} metric, say g, on a neighbourhood of the neck in the connected sum X_{I_{−}}#X_{I}. On the annulus Ω_{I} = ϕ_{I}({x ∈ ℝ^{4} : N^{−1}λ_{I}^{1/2} < |x| < Nλ_{I}^{1/2}}) we have

(3.36) (ϕ_{I}^{∗}f_{I}^{∗}g̃_{I})_{μν}(x) = \{ \begin{matrix}4λ_{I}^{2}(λ_{I}^{2} + |x|^{2})^{−2}(ϕ_{I}^{∗}g̃_{0})_{μν}(x) & if I_{−} = 0 \\ 4λ_{I}^{2}(λ_{I}^{2} + |x|^{2})^{−2}h_{1}^{−2}(x + q_{I})(ϕ_{I}^{∗}g̃_{I_{−}})_{μν}(x) & if I_{−} > 0. \end{matrix}

By comparing f_{I}^{∗}g̃_{I} and g_{I_{−}} on Ω_{I}, a little experimentation reveals that the C^{∞} conformal factors m_{I_{−}} and m_{I} can be chosen so that

(3.37)  κ^{−1} ≤ m_{I_{−}} ≤ κN^{4} on Ω_{I}(N^{−1}λ_{I}^{1/2}, Nλ_{I}^{1/2}),
κ^{−1} ≤ m_{I_{−}} ≤ κ on Ω_{I}(\frac{1}{2}λ_{I}^{1/2}, Nλ_{I}^{1/2}),
m_{I_{−}} = 1 on Ω_{I}(2λ_{I}^{1/2}, 4Nλ_{I}^{1/2}),

and likewise for m_{I} on Ω_{Is}, for some constant κ = κ(g_{0}). For each summand X_{I}, we smoothly extend the m_{I} to X′_{I} by setting m_{I} ≡ 1 away from the neck regions. This gives a C^{∞} metric g on X = #_{I∈ℐ}X_{I} by setting

(3.38)  g ≡ m_{I} g̃_{I} on X′_{I}, for all I ∈ ℐ.

The construction ensures that each m_{I} obeys

(3.39) κ^{−1} ≤ m_{I} ≤ κN^{4} on X′_{I}, κ^{−1} ≤ m_{I} ≤ κ on X′′_{I}, and m_{I} = 1 on X′′′_{I}.

Thus, the metrics g̃_{I} and g are equivalent on X′′_{I} with constants independent of N, and equivalent over X′_{I} with constants now depending on N.

The Hodge star operator ∗_{g} : Ω^{2}(X, ad P) → Ω^{2}(X, ad P) only depends on the conformal class [g] of g and so over each summand X′_{I} of X we have ∗_{g} = ∗_{m_{I} g̃_{I}} = ∗g̃_{I}. From Lemma 3.13, we obtain:

Lemma 3.14. There is a constant c = c(g_{0}) such that for any ζ ∈ Ω^{2}(X, ad P), then (a) ∥\frac{∂∗_{g}}{∂λ_{I}} ζ∥_{L^{∞}(X,g)} ≤ cNλ_{I}^{−1/2}∥ζ∥_{L^{∞}(X,g)},

(b) ∥\frac{∂∗_{g}}{∂p_{I}} ζ∥_{L^{∞}(X,g)} ≤ cNλ_{i_{1}}^{1/2}∥ζ∥_{L^{∞}(X,g)}.

We will often need to compare L^{p} norms defined by the different metrics g_{I}, g̃_{I}, and g over X′_{I} ⊂ X. The required “comparison estimates” are below follow in a straightforward way from Lemma 3.12 and Eq. (3.39), and similar inequalities may be found in [D-K, p.

Lemma 3.15. For any I ≥ 0, the following holds.

(a) If 2 ≤ p < ∞ and 4 ≤ q < ∞, there is a constant c = c(g_{0}, k, p, q), 1 ≤ c < ∞, such that for any ω ∈ Ω^{1}(X′_{I}, ad P_{I}) and ζ ∈ Ω^{2}(X′_{I}, ad P_{I}), then

∥ω∥_{L^{q}(X′_{I},g)} ≤ c∥ω∥_{L^{q}(X′_{I},g_{I})} and ∥ζ∥_{L^{p}(X′_{I},g)} ≤ c∥ζ∥_{L^{p}(X′_{I},g_{I})},
∥ω∥_{L^{q}(X′′_{I},g_{I})} ≤ c^{−1}∥ω∥_{L^{q}(X′′_{I},g)} and ∥ζ∥_{L^{p}(X′′_{I},g_{I})} ≤ c^{−1}∥ζ∥_{L^{p}(X′′_{I},g)}.

(b) If 1 ≤ p < ∞, n ≥ 1, and b_{0} is sufficiently small, there is a constant c = c(g_{0}, k, n, N, p), 1 ≤ c < ∞, such that for any α ∈ Ω^{n}(X′_{I}, ad P_{I}), then

c^{−1}∥α∥_{L^{p}(X′_{I},g_{I})} ≤ ∥α∥_{L^{p}(X′_{I},g̃_{I})}, ∥α∥_{L^{p}(X′_{I},g)} ≤ c∥α∥_{L^{p}(X′_{I},g_{I})}.

Lastly, having defined the conformal structure [g] of X, we apply the estimates for dψ_{I} in Eq. (3.22), the estimates for A′_{I} and F (A′_{I})^{+,g_{I}} in Lemma 3.9, and the estimates for ∗_{g} − ∗_{g_{I}} in Lemma 3.12 to obtain a bound for the L^{p}-norm of the g-self-dual curvature F^{+,g}(A′) = \frac{1}{2}(1 + ∗_{g})F (A′) of the connection A′ on the connected sum bundle P over X. Similar estimates have been given by Taubes and Donaldson.

Proposition 3.16. For 1 ≤ p < ∞ and sufficently small b_{0}, there exists a constant C = C(g_{0}, p, 𝒯) such that for any t ∈ 𝒯 one has ∥F^{+,g}(A′)∥_{L^{p}(X,g)} ≤ C\overline{b}^{4/p}.

3.6. Estimates over connected sums and conformal vector fields. The goal of this section is to obtain L^{2} estimates for the derivatives with respect to the scales λ_{I} and centres x_{I} of ad P̂-valued one-forms ω̂ over the base manifold X_{0} obtained by pulling back ad P -valued one-forms ω over the connected sum X.

Following Taubes [T84b], [T88], let us begin by defining some useful Sobolev norms on Ω^{1}(𝕊^{4}, ad P) and examine their behaviour under conformal diffeomorphisms. Suppose A is a C^{∞} connection on a G bundle P over 𝕊^{4}. Let g_{1} be the standard round metric on 𝕊^{4} and let δ be the flat metric on 𝕊^{4}\{s} obtained via the conformal identification ϕ_{n}^{−1} : 𝕊^{4}\{s} → ℝ^{4}. Let ∇^{A,g_{1}} denote the covariant derivative on Ω^{1}(𝕊^{4}, ad P) defined by the connection A and metric g_{1}, while ∇^{A,δ} denotes the covariant derivative on Ω^{1}(𝕊^{4} \ {s}, ad P) defined by A and δ. Define an L_{1}^{2} norm on Ω^{1}(𝕊^{4}, ad P) by

(3.40)  ∥ω∥_{L_{1}^{2}(𝕊^{4},A,g_{1})} ≡ ∥ω∥_{L^{2}(𝕊^{4},g_{1})} + ∥∇^{A,g_{1}}ω∥_{L_{1}^{2}(𝕊^{4},g_{1})}.

Similarly, if ω has compact support in 𝕊^{4} \ {s}, define

|ω|_{A} ≡ ∥∇^{A,δ}ω∥_{L^{2}(𝕊^{4},δ)}
(3.41)  ∥ω∥_{L_{1}^{2}(𝕊^{4},A,δ)} ≡ ∥ω∥_{L^{2}(𝕊^{4},δ)} + ∥∇^{A,δ}ω∥_{L^{2}(𝕊^{4},δ)}.

The properties of | · |_{A} and ∥ · ∥_{L_{1}^{2}(𝕊^{4},A,δ)} are described by the following result of [T84b]. Recall that C = D × T × SO(4) is identified, using ϕ_{n} : ℝ^{4} → 𝕊^{4} \ {s}, with the subgroup of conformal diffeomorphisms of (𝕊^{4}, g_{1}) which fix the south pole.

Lemma 3.17. [T84b, Proposition 2.4] Given an L_{1}^{2} connection A on a G bundle P over 𝕊^{4}, then the following holds.

(a) | · |_{A} extends to a continuous norm on L_{1}^{2}Ω^{1}(𝕊^{4}, ad P).

(b) The norm | · |_{A} is C-invariant: for any f ∈ C, |f^{∗}ω|_{f^{∗}A} = |ω|_{A}.

(c) There exists a constant 1 ≤ z < ∞, which is independent of P, A, f, and ω ∈ Ω^{1}(𝕊^{4}, ad P), such that

z^{−1}∥ω∥_{L_{1}^{2}(𝕊^{4},A,g_{1})} ≤ |ω|_{A} ≤ z∥ω∥_{L_{1}^{2}(𝕊^{4},A,g_{1})},
z^{−1}∥ω∥_{L_{1}^{2}(𝕊^{4},A,g_{1})} ≤ ∥ω∥_{L_{1}^{2}(𝕊^{4},A,δ)} ≤ z∥ω∥_{L_{1}^{2}(𝕊^{4},A,g_{1})}.

Lemma 3.18. [T88, Lemma 3.1] Let A be a C^{∞} connection on a G bundle P over 𝕊^{4} with its standard metric g_{1} and let f : 𝕊^{4} → 𝕊^{4} be a conformal diffeomorphism. Then there exists a constant 1 ≤ z < ∞, which is independent of P, A, f, and ω ∈ Ω^{1}(𝕊^{4}, ad P), with the following significance:

z^{−1}∥ω∥_{L_{1}^{2}(𝕊^{4},A,g_{1})} ≤ ∥f^{∗}ω∥_{L_{1}^{2}(𝕊^{4},f^{∗}A,g_{1})} ≤ z∥ω∥_{L_{1}^{2}(𝕊^{4},A,g_{1})}.

Recall that c_{λ} denotes both the dilation x ↦ x/λ of ℝ^{4} and the conformal diffeomorphism of (𝕊^{4}, g_{1}) induced by ϕ_{n}. A straightforward application of Hölder’s inequality yields the following “transfer estimates” for the maps c_{λ}.

Lemma 3.19. Let 2 ≤ p ≤ p_{1} ≤ 4, let λ ∈ (0, 1], and let U be an open subset of 𝕊^{4} \ B(s, Nλ^{1/2}). Let P be a G bundle over 𝕊^{4}. Then there is a constant C = C(N) such that the following holds.

(a) If ω ∈ Ω^{1}(U, ad P), then ∥c_{λ}^{∗}ω∥_{L^{p}(c_{λ}^{−1}(U),g_{1})} ≤ Cλ^{2/p−2/p_{1}}∥ω∥_{L^{p_{1}} (U,g_{1})}.

(b) If ζ ∈ Ω^{2}(U, ad P), then ∥c_{λ}^{∗}ζ∥_{L^{2}(c_{λ}^{−1}(U),g_{1})} ≤ C∥ζ∥_{L^{2}(U,g_{1})}.

We next consider the action of the conformal group on Ω^{1}(𝕊^{4}, ad P). Let f_{λ,q} denote the lift to 𝕊^{4}, via the chart ϕ_{n}, of the conformal diffeomorphism c_{λ} ∘ τ_{q} on ℝ^{4}. Let P be a G bundle over 𝕊^{4} and suppose ω ∈ Ω^{1}(𝕊^{4}, ad P). Then Eq. (3.8) gives

(3.42)  \frac{∂f_{λ,q}^{∗}ω}{∂λ} = −\frac{1}{λ}f_{λ,q}^{∗}ℒ_{r}ω and \frac{∂f_{λ,q}^{∗}ω}{∂p} = −\frac{1}{λ}f_{λ,q}^{∗}ℒ_{p}ω,

where ∂/∂p ≡ p^{μ}∂/∂q^{μ}. It will be convenient to express the above Lie derivatives in terms of covariant derivatives. If A is a C^{∞} connection on P, then Eqs. (3.6) and (3.7) imply

(3.43)  ℒ_{r}ω = ω + ∇_{r}^{A,δ}ω and ℒ_{p}ω = ∇_{p}^{A,δ}ω.

This leads to the following estimates for the derivatives of f_{λ,q}^{∗}ω with respect to λ and q.

Lemma 3.20. Let A be a C^{∞} connection on a G bundle P over 𝕊^{4}, let U ⊂ 𝕊^{4} \ B(s, Nλ^{1/2}) be an open subset, and let ω ∈ Ω^{1}(U, ad P). Let ∂/∂p = p^{μ}∂/∂q^{μ}, |p| ≤ 1. Then there is a constant C = C(q, N) such that the following bounds hold.

(a) ∥∂f_{λ,q}^{∗}ω/∂λ∥_{L^{2}(f_{λ,q}^{−1}(U),g_{1})} ≤ Cλ^{−1/2}∥ω∥_{L_{1}^{2}(U,A,g_{1})};

(b) ∥∂f_{λ,q}^{∗}ω/∂p∥_{L^{2}(f_{λ,q}^{−1}(U),g_{1})} ≤ C∥ω∥_{L_{1}^{2}(U,A,g_{1})}.

Proof. (a) Observe that U = ϕ_{n}(B(0, Nλ^{−1/2})) and f_{λ,q}^{−1}(U) = ϕ_{n}(B(q, Nλ^{1/2})). From Eqs. (3.42) and (3.43), we have

\frac{∂f_{λ,q}^{∗}ω}{∂λ} = −λ^{−1}f_{λ,q}^{∗}ℒ_{r}ω and f_{λ,q}^{∗}ℒ_{r}ω = f_{λ,q,∗}^{−1}r⌟f_{λ,q}^{∗}∇^{A,δ}ω + f_{λ,q}^{∗}ω on f_{λ,q}^{−1}(U),

where r = y^{μ}∂/∂y^{μ} and f_{λ,q,∗}^{−1}r = x^{μ}∂/∂x^{μ} with respect to the coordinates y = ϕ_{n}^{−1} on U and x = τ_{q} ∘ ϕ_{n}^{−1} on f_{λ,q}^{−1}(U). Since |f_{λ,q,∗}^{−1}r|_{g_{1}} ≤ Cλ^{1/2} on f_{λ,q}^{−1}(U), Lemma 3.19 implies

∥f_{λ,q}^{∗}ℒ_{r}ω∥_{L^{2}(f_{λ,q}^{−1}(U),g_{1})} ≤ ∥f_{λ,q}^{∗}ω∥_{L^{2}(f_{λ,q}^{−1}(U),g_{1})} + Cλ^{1/2}∥f_{λ,q}^{∗}∇^{A,δ}ω∥_{L^{2}(f_{λ,q}^{−1}(U),g_{1})}
≤ Cλ^{1/2}∥ω∥_{L^{4}(U,g_{1})} + Cλ^{1/2}∥∇^{A,δ}ω∥_{L^{2}(U,g_{1})}
= Cλ^{1/2}∥ω∥_{L_{1}^{2}(U,A,δ)},

the last step following by conformal invariance. Lemma 3.17 then gives (a).

(b) From Eqs. (3.42) and (3.43), we have

\frac{∂f_{λ,q}^{∗}ω}{∂p_{I}} = −λ^{−1}f_{λ,q}^{∗}ℒ_{p}ω and f_{λ,q}^{∗}ℒ_{p}ω = f_{λ,q,∗}^{−1}p⌟f_{λ,q}^{∗}∇^{A,δ}ω on f_{λ,q}^{−1}(U),

where p = p^{μ}∂/∂y^{μ} on U and f_{λ,q,∗}^{−1}p = λp^{μ}∂/∂x^{μ} on f_{λ,q}^{−1}(U). Since |f_{λ,q,∗}^{−1}p|_{g_{1}} ≤ Cλ on f_{λ,q}^{−1}(U), Lemma 3.19 implies

∥f_{λ,q}^{∗}ℒ_{p}ω∥_{L^{2}(f_{λ,q}^{−1}(U),g_{1})} ≤ Cλ∥f_{λ,q}^{∗}∇^{A,δ}ω∥_{L^{2}(f_{λ,q}^{−1}(U),g_{1})}
≤ Cλ∥∇^{A,δ}ω∥_{L^{2}(U,δ)} ≤ Cλ∥ω∥_{L_{1}^{2}(U,A,δ)}.

Lemma 3.17 then gives (b). □

We will frequently need to compute estimates for families of one-forms ω over connected sums X and to this end, it will be useful to define suitable Sobolev norms which depend only on the fixed connections A_{I} and, in particular, the fixed metrics g_{I} on each summand X_{I} rather than varying metric g on X. Let P be the G bundle over the connected sum X = #_{I∈ℐ}X_{I} defined in §3.3. Then we may view any ω ∈ Ω^{1}(X, ad P) as a collection of ω_{I} ∈ Ω^{1}(X′_{I}, ad P_{I}) which agree over the necks Ω_{I} = f_{I}^{−1}(Ω_{Is}) connecting each pair X_{I_{−}} and X_{I}:

σ_{I}^{∗}ω_{I_{−}} = Ad (ρ_{I}^{−1})f_{I}^{∗}σ_{Is}^{∗}ω_{I} on Ω_{I},

where f_{I} : Ω_{I} → Ω_{Is} is the identification map.

From §3.5, we recall that there is a C^{∞} metric g on X which agrees, modulo the conformal factors m_{I}, with the metrics g_{0} on the base X′_{0} and g̃_{I} ≃ g_{1} on the four-spheres X′_{I}. Moreover, the L^{q} norms on Ω^{1}(X′_{I}, ad P_{I}), 4 ≤ q < ∞, and L^{p} norms on Ω^{2}(X′_{I}, ad P_{I}), 2 ≤ p < ∞, compare uniformly when defined with the metrics g_{I}, g̃_{I}, or g = m_{I} g̃_{I} on X′_{I}. The constants involved in these norm comparisons are independent of the scale parameters λ_{J} for forms supported on X′_{I} and independent of both the λ_{J} and N for forms supported on X′′_{I}. Thus, we may conveniently define L^{q} norms on Ω^{1}(X, ad P), 4 ≤ q < ∞, and L^{p} norms on Ω^{2}(X, ad P), 2 ≤ p < ∞, using the metric g on X.

In Chapter 5, we will need to bound the L_{1}^{2} norms of solutions ω ∈ Ω^{1}(X, ad P) to the g-anti-self-dual equation F^{+,g}(A′ + ω) = 0 over X. Unfortunately, since the conformal factors m_{I} have badly behaved derivatives over the neck regions, the norm comparisons described above do not hold for L_{n}^{2} Sobolev norms if n ≥ 1. Of course, problems of this type are encountered in [D86], [D-K], and [T92]. So, given such an ω ∈ Ω^{1}(X, ad P), with ω = {ω_{I}}_{I∈ℐ} as above, and 1 ≤ p < ∞, define

(3.44)  ∥ω∥_{ℒ_{1}^{p}(X)} ≡ ∑_{I∈ℐ}∥ω_{I}∥_{L_{1}^{p}(X_{I},A_{I},g_{I})},

by analogy with Eq. (6.25) in [T92].

Recall that a one-form ω ∈ Ω^{1}(X, ad P) pulls back to a one-form ω̂ ∈ Ω^{1}(X_{0}, ad P̂) defined by

(3.45)  ω̂ = f_{0}^{∗} ⋯ f_{J}^{∗}ω on f_{0}^{−1} ⋯ f_{J}^{−1}(X′_{J}) ⊂ X_{0},

for each J ∈ ℐ. We will need estimates for the derivatives of ω̂ with respect to the scales λ_{I} and centres x_{I}. To begin, we need suitable expressions for these derivatives:

Lemma 3.21. Let ω ∈ Ω^{1}(X′_{J}, ad P_{J}), let 0 < I ≤ J, and let ∂/∂p_{I} = p_{I}^{μ}∂/∂q_{I}^{μ}. Then:

(a) \frac{∂}{∂λ_{I}} f_{0}^{∗} ⋯ f_{J}^{∗}ω = f_{0}^{∗} ⋯ f_{J}^{∗} \frac{∂ω}{∂λ_{I}}, for J < I;

(b) \frac{∂}{∂λ_{I}} f_{0}^{∗} ⋯ f_{I}^{∗}ω = −λ_{I}^{−1}f_{0}^{∗} ⋯ f_{I}^{∗}ℒ_{r}ω, for J = I;

(c) \frac{∂}{∂λ_{I}} f_{0}^{∗} ⋯ f_{J}^{∗}ω = −λ_{I}^{−1}f_{0}^{∗} ⋯ f_{I}^{∗}ℒ_{r}f_{I_{+}}^{∗} ⋯ f_{J}^{∗}ω, for J > I;

(d) \frac{∂}{∂p_{I}} f_{0}^{∗} ⋯ f_{J}^{∗}ω = f_{0}^{∗} ⋯ f_{J}^{∗} \frac{∂ω}{∂p_{I}}, for J < I;

(e) \frac{∂}{∂p_{I}} f_{0}^{∗} ⋯ f_{I}^{∗}ω = −λ_{I}^{−1}f_{0}^{∗} ⋯ f_{I}^{∗}ℒ_{p}ω, for J = I;

(f) \frac{∂}{∂p_{I}} f_{0}^{∗} ⋯ f_{J}^{∗}ω = −λ_{I}^{−1}f_{0}^{∗} ⋯ f_{I}^{∗}ℒ_{p}f_{I_{+}}^{∗} ⋯ f_{J}^{∗}ω, for J > I.

Remark 3.22. When I_{−} = 0, then ∂/∂p_{I} = p_{I}^{μ}∂/∂p_{I}^{μ} and f_{I} = ϕ_{In} ∘ c_{I} ∘ ϕ_{I}^{−1} is replaced by \overline{f}_{I} = ϕ_{In} ∘ c_{I} ∘ τ_{p_{I}} ∘ ϕ_{I}^{−1} in order to compute the derivative at p_{I} = 0.

These expressions lead to the following bounds for the derivatives with respect to the scales λ_{I} and centres x_{I} of the pull-backs f_{0}^{∗} ⋯ f_{J}^{∗}ω.

Lemma 3.23. Let ω ∈ Ω^{1}(X′_{J}, ad P_{J}), let U = f_{0}^{−1} ∘ ⋯ ∘ f_{J}^{−1}(X′_{J}) ⊂ X_{0}, let 0 < I ≤ J, and let ∂/∂p_{I} = p_{I}^{μ}∂/∂q_{I}^{μ} with |p_{I}| ≤ 1. Then there is a constant C = C(g_{0}, N) such that the following holds.

(a) ∥\frac{∂}{∂λ_{I}} f_{0}^{∗} ⋯ f_{J}^{∗}ω∥_{L^{2}(U,g_{0})} ≤ C ∥\frac{∂ω}{∂λ_{I}} ∥_{L^{2}(X′_{J} ,g_{J})}, for J < I;

(b) ∥\frac{∂}{∂λ_{I}} f_{0}^{∗} ⋯ f_{J}^{∗}ω∥_{L^{2}(U,g_{0})} ≤ Cλ_{I}^{−1/2}∥ω∥_{L_{1}^{2}(X′_{J} ,A_{J} ,g_{J})}, for J ≥ I;

(c) ∥\frac{∂}{∂p_{I}} f_{0}^{∗} ⋯ f_{J}^{∗}ω∥_{L^{2}(U,g_{0})} ≤ C ∥\frac{∂ω}{∂p_{I}} ∥_{L^{2}(X′_{J} ,g_{J})}, for J < I;

(d) ∥\frac{∂}{∂p_{I}} f_{0}^{∗} ⋯ f_{J}^{∗}ω∥_{L^{2}(U,g_{0})} ≤ C∥ω∥_{L_{1}^{2}(X′_{J} ,A_{J} ,g_{J})}, for J ≥ I.

Proof. (a) By repeatedly applying Lemma 3.19, we find that

∥\frac{∂}{∂λ_{I}} f_{0}^{∗} ⋯ f_{J}^{∗}ω∥_{L^{2}(U,g_{0})} = ∥f_{0}^{∗} ⋯ f_{J}^{∗} \frac{∂ω}{∂λ_{I}} ∥_{L^{2}(U,g_{0})} ≤ C ∥\frac{∂ω}{∂λ_{I}} ∥_{L^{2}(X′_{J} ,g_{J})} ,

as required for (a). For J = I and U = f_{0}^{−1} ∘ ⋯ ∘ f_{I}^{−1}(X′_{I}) ⊂ X_{0}, Lemmas 3.19 and 3.20 show that

∥\frac{∂}{∂λ_{I}} f_{0}^{∗} ⋯ f_{I}^{∗}ω∥_{L^{2}(U,g_{0})} = ∥f_{0}^{∗} ⋯ f_{I_{−}}^{∗} \frac{∂f_{I}^{∗}ω}{∂λ_{I}} ∥_{L^{2}(U,g_{0})}

≤ C ∥\frac{∂f_{I}^{∗}ω}{∂λ_{I}} ∥_{L^{2}(f_{I}^{−1}(X′_{I}),g_{I_{−}})} ≤ Cλ_{I}^{−1/2}∥ω∥_{L_{1}^{2}(X′_{I},A_{I},g_{I})}.

Let V = f_{I_{+}}^{−1} ∘ ⋯ ∘ f_{J}^{−1}(X′_{J}) ⊂ X′_{I}, so that U = f_{0}^{−1} ∘ ⋯ ∘ f_{I}^{−1}(V) ⊂ X_{0}. Then for J > I, we have

∥\frac{∂}{∂λ_{I}} f_{0}^{∗} ⋯ f_{J}^{∗}ω∥_{L^{2}(U,g_{0})} = ∥f_{0}^{∗} ⋯ f_{I_{−}}^{∗} \frac{∂}{∂λ_{I}} f_{I}^{∗} ⋯ f_{J}^{∗}ω∥_{L^{2}(U,g_{0})}

≤ C ∥\frac{∂}{∂λ_{I}} f_{I}^{∗}f_{I_{+}}^{∗} ⋯ f_{J}^{∗}ω∥_{L^{2}(f_{I}^{−1}(V),g_{I_{−}})}

≤ Cλ_{I}^{−1/2}∥f_{I_{+}}^{∗} ⋯ f_{J}^{∗}ω∥_{L_{1}^{2}(V,f_{I_{+}}^{∗} ···f_{J}^{∗}A_{J} ,g_{I})}

≤ C∥ω∥_{L_{1}^{2}(X′_{J} ,A_{J} ,g_{J})}.

by repeatedly applying Lemma 3.18 in the last step. This gives (b); the proofs of (c) and (d) are similar. □

Finally, we obtain our estimate for the derivatives of ω̂ with respect to the scales λ_{I} and centres x_{I}.

Proposition 3.24. There is a constant C = (g_{0}, 𝒯) such that for any ω ∈ Ω^{1}(X, ad P) and t ∈ 𝒯, the following bounds hold.

(a) ∥∂ω̂/∂λ_{I}∥_{L^{2}(X_{0},g_{0})} ≤ C(∥∂ω/∂λ_{I}∥_{L^{2}(X,g)} + λ_{I}^{−1/2}∥ω∥_{ℒ_{1}^{2}(X)} );

(b) ∥∂ω̂/∂p_{I}∥_{L^{2}(X_{0},g_{0})} ≤ C(∥∂ω/∂p_{I}∥_{L^{2}(X,g)} + ∥ω∥_{ℒ_{1}^{2}(X)} ).

Proof. From Lemma 3.23 we have

∥\frac{∂ω̂}{∂λ_{I}} ∥_{L^{2}(X_{0},g_{0})} ≤ C ∑_{J<I} ∥\frac{∂ω}{∂λ_{I}} ∥_{L^{2}(X′_{J} ,g_{J})} + Cλ_{I}^{−1/2} ∑_{J≥I} ∥ω∥_{L_{1}^{2}(X′_{J} ,A_{J} ,g_{J})},

and so (a) follows from Lemma 3.15. Similarly, Lemma 3.23 gives

∥\frac{∂ω̂}{∂p_{I}} ∥_{L^{2}(X_{0},g_{0})} ≤ C ∑_{J<I} ∥\frac{∂ω}{∂p_{I}} ∥_{L^{2}(X′_{J} ,g_{J})} + C ∑_{J≥I} ∥ω∥_{L_{1}^{2}(X′_{J} ,A_{J} ,g_{J})}.

and likewise, (b) follows from Lemma 3.15. □

3.7. Derivatives with respect to scales and centres. We obtain L^{p} estimates for the derivatives of the connections A′ and Â′ and of the g-self-dual curvature F^{+,g}(A′) with respect to the scales λ_{I} and centres x_{I}.

Throughout this section we require that b_{J} = 4Nλ_{J}^{1/2} for all J. Let us first record the following bounds for the derivatives of the cutoff functions ψ_{J} for J = I_{−} or I:

(3.46) |∂ψ_{J}/∂λ_{I}|_{g_{J}} ≤ CN^{−1}λ_{I}^{−1} and |∂dψ_{J}/∂λ_{I}|_{g_{J}} ≤ CN^{−2}λ_{I}^{−3/2} on X′_{J},
|∂ψ_{J}/∂p_{I}|_{g_{J}} ≤ CN^{−1}λ_{I}^{−1/2} and |∂dψ_{J}/∂p_{I}|_{g_{J}} ≤ CN^{−2}λ_{I}^{−1} on X′_{J},

where ∂/∂p_{I} ≡ p_{I}^{μ}∂/∂q_{I}^{μ} and |p_{I}| ≤ 1. The constant C depends only on g_{J}. We now begin with the L^{p} estimates for derivatives of the connections A′.

Proposition 3.25. Suppose 1 ≤ p < ∞ and I > 0. Then for sufficiently small λ_{0}, there is a constant C = C(g_{0}, p, 𝒯) such that for any t ∈ 𝒯,

(a) ∥∂A′/∂λ_{I}∥_{L^{p}(X,g)} ≤ Cλ_{I}^{2/p−1/2},

(b) ∥∂A′/∂p_{I}∥_{L^{p}(X,g)} ≤ Cλ_{I}^{2/p}.

Proof. (a) Observe that ∂A′/∂λ_{I} is non-zero only on the supports of ∂ψ_{I_{−}}/∂λ_{I} and ∂ψ_{I}/∂λ_{I}, given by the annuli Ω_{I}(\frac{1}{2}b_{I}, b_{I}) in X′_{I_{−}} and Ω_{Is}(\frac{1}{2}b_{I}, b_{I}) in X′_{I}.

Step 1. Estimate of ∂A′/∂λ_{I} over X′_{I_{−}}. Recall that ψ_{I_{−}} = 1 on the complement of the balls B_{I}(b_{I}) in X_{I_{−}}, while 0 < ψ_{I_{−}} < 1 on Ω_{I}(\frac{1}{2}b_{I}, b_{I}), and ψ_{I_{−}} = 0 on B_{I}(\frac{1}{2}b_{I}).

We have σ_{I}^{∗}A′ = ψ_{I_{−}}σ_{I}^{∗}A_{I_{−}} on Ω_{I}(\frac{1}{2}b_{I}, b_{I}) and thus σ_{I}^{∗} \frac{∂A′}{∂λ_{I}} = \frac{∂ψ_{I_{−}}}{∂λ_{I}} σ_{I}^{∗}A_{I_{−}} on X′_{I_{−}}. Since

|∂ψ_{I}/∂λ_{I}| ≤ Cλ_{I}^{−1} by Eq. (3.46) and |σ_{I}^{∗}A_{I_{−}}|_{g_{I_{−}}} ≤ Cλ_{I}^{1/2} on Ω_{I}(\frac{1}{2}b_{I}, b_{I}) by Lemmas 3.1 and 3.3, we obtain the pointwise bound

|\frac{∂A′}{∂λ_{I}} |_{g_{I_{−}}} ≤ \{ \begin{matrix}Cλ_{I}^{−1/2} & on Ω_{I}(\frac{1}{2}b_{I}, b_{I}) \\ 0 & on X′_{I_{−}} \ Ω_{I}(\frac{1}{2}b_{I}, b_{I}). \end{matrix}

Hence, we get the integral estimate

(3.47)  ∫_{X′_{I_{−}}} |\frac{∂A′}{∂λ_{I}} |_{g}^{p} dV_{g} ≤ Cλ_{I}^{2−p/2},

noting that g = g̃_{I_{−}} on X_{I_{−}} \ B_{I}(\frac{1}{2}b_{I}) and appealing to Lemma 3.12. Step 2. Estimate of ∂A′/∂λ_{I} over X′_{I}. A similar argument shows that

(3.48)  ∫_{X′_{I}} |\frac{∂A′}{∂λ_{I}} |_{g}^{p} dV_{g} ≤ Cλ_{I}^{2−p/2},

and combining the integral bounds from Steps 1 and 2 gives (a). For (b) we use the pointwise estimates |∂ψ_{J}/∂p_{I}| ≤ Cλ_{I}^{−1/2}, J = I_{−}, I. The same argument as in (a) then gives the required bound. □

Our next task is to obtain a L^{p} estimates for the derivatives of the g-self-dual curvature F^{+,g}(A′).

Proposition 3.26. Suppose 1 ≤ p < 4 and I > 0. Then for sufficiently small λ_{0}, there exists a constant C = C(g_{0}, p, 𝒯) such that for any t ∈ 𝒯,

(a) ∥∂F^{+,g}(A′)/∂λ_{I}∥_{L^{p}(X,g)} ≤ Cλ_{I}^{2/p−1},

(b) ∥∂F^{+,g}(A′)/∂p_{I}∥_{L^{p}(X,g)} ≤ C(λ_{I}^{2/p−1/2} + \overline{λ}^{1/2p}).

Proof. (a) We note that F^{+,g}(A′) = F^{+,g̃_{J}} (ψ_{J}A_{J}) on X′_{J} and so ∂F^{+,g}(A′)/∂λ_{I} is supported on ∪_{J≥I_{−}}X′_{J}. It is convenient to obtain estimates separately over the regions X′_{I_{−}}, X′_{I}, and X′_{J}, J > I. Step 1. Estimate of ∂F^{+,g}(A′)/∂λ_{I} over X′_{I_{−}}. On the annulus Ω_{I}(\frac{1}{2}b_{I}, b_{I}) we have F^{+,g}(A′) = \frac{1}{2}(1 + ∗g̃_{I_{−}})F (ψ_{I_{−}}A_{I_{−}}) and

F (ψ_{I_{−}}A_{I_{−}}) = ψ_{I_{−}}F (A_{I_{−}}) + dψ_{I_{−}} ∧ σ_{I}^{∗}A_{I_{−}} + (ψ_{I_{−}}^{2} − ψ_{I_{−}})σ_{I}^{∗}A_{I_{−}} ∧ σ_{I}^{∗}A_{I_{−}}.

Therefore, we see that

\frac{∂F^{+,g}(A′)}{∂λ_{I}} = \frac{1}{2}(1 + ∗g̃_{I_{−}})\frac{∂F (ψ_{I_{−}}A_{I_{−}})}{∂λ_{I}},
\frac{∂F (ψ_{I_{−}}A_{I_{−}})}{∂λ_{I}} = \frac{∂ψ_{I_{−}}}{∂λ_{I}} F (A_{I_{−}}) + \frac{∂dψ_{I_{−}}}{∂λ_{I}} ∧ σ_{I}^{∗}A_{I_{−}} + (2ψ_{I_{−}} − 1)\frac{∂ψ_{I_{−}}}{∂λ_{I}} σ_{I}^{∗}A_{I_{−}} ∧ σ_{I}^{∗}A_{I_{−}}.

on X′_{I_{−}}. The metric g̃_{I_{−}} is independent of λ_{I} and so applying the pointwise estimates of Lemmas 3.1, 3.3, and Eq. (3.46), we find that

|\frac{∂F^{+,g}(A′)}{∂λ_{I}} |_{g_{I_{−}}} ≤ \{ \begin{matrix}Cλ_{I}^{−1} & on Ω_{I}(\frac{1}{2}b_{I}, b_{I}) \\ 0 & on X_{I_{−}} \ Ω_{I}(\frac{1}{2}b_{I}, b_{I}). \end{matrix}

Consequently, we obtain

(3.49)  ∫_{X′_{I_{−}}} |\frac{∂F^{+,g}(A′)}{∂λ_{I}} |_{g}^{p} dV_{g} ≤ Cλ_{I}^{2−p},

where we observe that g = g̃_{I_{−}} on Ω_{I}(\frac{1}{2}b_{I}, b_{I}).

Step 2. Estimate of ∂F^{+,g}(A′)/∂λ_{I} over X′_{I}. We have F^{+,g}(A′) = \frac{1}{2}(1 + ∗g̃_{I})F (ψ_{I}A_{I}) and F (ψ_{I}A_{I}) = ψ_{I}F (A_{I}) + dψ_{I} ∧ σ_{Is}^{∗}A_{I} + (ψ_{I}^{2} − ψ_{I})σ_{Is}^{∗}A_{I} ∧ σ_{Is}^{∗}A_{I} on X′_{I}. Thus,

\frac{∂F^{+,g}(A′)}{∂λ_{I}} = \frac{1}{2} \frac{∂∗g̃_{I}}{∂λ_{I}} F (ψ_{I}A_{I}) + \frac{1}{2}(1 + ∗g̃_{I})\frac{∂F (ψ_{I}A_{I})}{∂λ_{I}},
\frac{∂F (ψ_{I}A_{I})}{∂λ_{I}} = \frac{∂ψ_{I}}{∂λ_{I}} F (A_{I}) + \frac{∂dψ_{I}}{∂λ_{I}} ∧ σ_{Is}^{∗}A_{I} + (2ψ_{I} − 1)\frac{∂ψ_{I}}{∂λ_{I}} σ_{Is}^{∗}A_{I} ∧ σ_{Is}^{∗}A_{I}

on X′_{I}. Applying the pointwise estimates of Lemmas 3.3, 3.12, 3.13, and Eq. (3.46), we find that

|ϕ_{In}^{∗} \frac{∂F^{+,g}(A′)}{∂λ_{I}} |_{g_{I}} (x) ≤ \{ \begin{matrix}0 & on B_{Is}(\frac{1}{2}b_{I}) \\ Cλ_{I}^{−1} & on Ω_{Is}(\frac{1}{2}b_{I}, b_{I}) \\ C|x| & on X_{I} \ B_{Is}(b_{I}). \end{matrix}

Now g = g̃_{I} on X_{I} \B_{Is}(\frac{1}{2}b_{I}), and so applying the above estimates and Hölder’s inequality gives

(3.50)  ∫_{X′_{I}} |\frac{∂F^{+,g}(A′)}{∂λ_{I}} |_{g}^{p} dV_{g} ≤ Cλ_{I}^{2−p},

completing Step 2. Step 3. Estimate of ∂F^{+,g}(A′)/∂λ_{I} over X′_{J}, J > I. We have

\frac{∂F^{+,g}(ψ_{J}A_{J})}{∂λ_{I}} = \frac{1}{2} \frac{∂∗g̃_{J}}{∂λ_{I}} F (ψ_{J}A_{J}) on X′_{J},

since F^{+,g}(A′) = \frac{1}{2}(1 + ∗g̃_{J})F (ψ_{J}A_{J}). The pointwise estimates of Lemmas 3.9, 3.12, and 3.13 show that

|ϕ_{Jn}^{∗} \frac{∂F^{+,g}(A′)}{∂λ_{I}} |_{g_{J}} (x) ≤ \{ \begin{matrix}0 & on B_{Js}(\frac{1}{2}b_{J}) \\ C|x| & on X_{J} \ B_{Js}(\frac{1}{2}b_{J}). \end{matrix}

Again, g = g̃_{J} on X_{J} \ B_{Js}(\frac{1}{2}b_{J}), and so

(3.51)  ∫_{X′_{J}} |\frac{∂F^{+,g}(A′)}{∂λ_{I}} |_{g}^{p} dV_{g} ≤ C.

Combining the integral estimates of Steps 1 to 3 then gives (a).

(b) The argument is the same, except that we now use the cutoff function estimates |∂ψ_{J}/∂p_{I}| ≤ Cλ_{I}^{−1/2}, |∂dψ_{J}/∂p_{I}| ≤ Cλ_{I}^{−1}, J = I_{−}, I, and metric estimates |∂g̃_{J}/∂p_{I}| ≤

CN\overline{λ}^{1/2}, J ≥ I. □

Lastly, we have L^{2} estimates of the derivatives of Â′ with respect to λ_{I} and x_{I}.

Proposition 3.27. Suppose I > 0. Then for sufficiently small λ_{0}, there is a constant C = C(g_{0}, 𝒯) such that for any t ∈ 𝒯,

(a) ∥∂Â′/∂λ_{I}∥_{L^{2}(X_{0},g_{0})} ≤ C,

(b) ∥∂Â′/∂p_{I}∥_{L^{2}(X_{0},g_{0})} ≤ C.

Proof. (a) The connection one-forms over X_{0} having non-zero derivatives with respect to λ_{I} are given by

Â′ = \{ \begin{matrix}f_{0}^{∗} ⋯ f_{I_{−}}^{∗}ψ_{I_{−}}A_{I_{−}} & over f_{0}^{−1} ∘ ⋯ ∘ f_{I_{−}}^{−1}(X′_{I_{−}}) ⊂ X_{0} \\ f_{0}^{∗} ⋯ f_{I}^{∗}Â′_{I} & over X_{I} \ B_{Is}(N_{I}^{−1}λ_{I}^{1/2}), \end{matrix}

where Â′_{I} is the C^{∞} connection over X_{I}, I > 0, given by

Â′_{I} = \{ \begin{matrix}f_{I_{+}}^{∗} ⋯ f_{J}^{∗}ψ_{J}A_{J} & over the regions f_{I_{+}}^{−1} ∘ ⋯ ∘ f_{J}^{−1}(X′_{J}) ⊂ X_{I} \\ ψ_{I}A_{I} & over the complement of these regions in X_{I}. \end{matrix}

It is convenient to consider the estimates over these different regions of X separately. Step 1. Estimate of ∂f_{0}^{∗} ⋯ f_{I_{−}}^{∗}ψ_{I_{−}}A_{I_{−}}/∂λ_{I}. We have Â′ = f_{0}^{∗} ⋯ f_{I_{−}}^{∗}ψ_{I_{−}}A_{I_{−}}, which is supported on U_{1} ≡ f_{0}^{−1} ∘ ⋯ ∘ f_{I_{−}}^{−1}(X′_{I_{−}}) ⊂ X_{0}, and so \frac{∂}{∂λ_{I}} Â′ = f_{0}^{∗} ⋯ f_{I_{−}}^{∗} \frac{∂}{∂λ_{I}} ψ_{I_{−}}A_{I_{−}} on U_{1}. Lemma 3.19 implies that

∥f_{0}^{∗} ⋯ f_{I_{−}}^{∗} \frac{∂ψ_{I_{−}}A_{I_{−}}}{∂λ_{I}} ∥_{L^{2}(U_{1},g_{0})} ≤ C ∥\frac{∂ψ_{I_{−}}A_{I_{−}}}{∂λ_{I}} ∥_{L^{2}(X′_{I_{−}} ,g_{I_{−}})} .

We have σ_{I}^{∗}ψ_{I_{−}}A_{I_{−}} = ψ_{I_{−}}σ_{I}^{∗}A_{I_{−}}, where the section σ_{I} is chosen so that σ_{I}^{∗}A_{I_{−}} is in radial gauge, and so the pointwise estimates of Lemmas 3.1, 3.3, and Eq. (3.46) show that

|\frac{∂ψ_{I_{−}}A_{I_{−}}}{∂λ_{I}} |_{g_{I_{−}}} ≤ \{ \begin{matrix}Cλ_{I}^{−1/2} & on Ω_{I}(\frac{1}{2}b_{I}, b_{I}) \\ 0 & on X_{I_{−}} \ B_{I}(b_{I}). \end{matrix}

Noting that g = g̃_{I_{−}} on X_{I_{−}} \ B_{I}(b_{I}), we obtain the integral bound

∫_{X′_{I_{−}}} |\frac{∂ψ_{I_{−}}A_{I_{−}}}{∂λ_{I}} |_{g}^{2} dV_{g} ≤ Cλ_{I},

and combining the preceding integral estimates gives ∥\frac{∂}{∂λ_{I}} Â′∥_{L^{2}(U_{1},g_{0})} ≤ Cλ_{I}^{1/2}, completing Step 1.

Step 2. Estimate of ∂f_{0}^{∗} ⋯ f_{I}^{∗}ψ_{I}A_{I}/∂λ_{I}. We denote Â′ = f_{0}^{∗} ⋯ f_{I}^{∗}ψ_{I}A_{I}, which is supported on U_{2} ≡ f_{0}^{−1} ∘ ⋯ ∘ f_{I}^{−1}(X′_{I}) ⊂ X_{0}, and so \frac{∂}{∂λ_{I}} Â′ = f_{0}^{∗} ⋯ f_{I_{−}}^{∗} \frac{∂}{∂λ_{I}} f_{I}^{∗}ψ_{I}A_{I} on U_{2}. Repeated application of Lemma 3.19 then gives the integral bound

∥f_{0}^{∗} ⋯ f_{I_{−}}^{∗} \frac{∂f_{I}^{∗}ψ_{I}A_{I}}{∂λ_{I}} ∥_{L^{2}(U_{2},g_{0})} ≤ C ∥\frac{∂f_{I}^{∗}ψ_{I}A_{I}}{∂λ_{I}} ∥_{L^{2}(X′_{I_{−}} ,g_{I_{−}})} .

Recall that Eq. (3.9) implies \frac{∂}{∂λ_{I}} f_{I}^{∗}ψ_{I}A_{I} = −λ_{I}^{−1}f_{I}^{∗}ι_{r}F (ψ_{I}A_{I}) on B′_{I}. The curvature

F (ψ_{I}A_{I}) is supported on X_{I} \ B_{Is}(\frac{1}{2}b_{I}) and \frac{∂}{∂λ_{I}} f_{I}^{∗}ψ_{I}A_{I} is supported on B_{I}(\frac{1}{2}N_{I}^{−1}λ_{I}^{1/2}). Then,

|ϕ_{In}^{∗}ι_{r}F (ψ_{I}A_{I})|_{δ_{I}} (x) ≤ K \frac{|x|}{(1 + |x|^{2})^{2}} ,

and since ϕ_{I}^{∗}f_{I}^{∗}ι_{r}F (ψ_{I}A_{I})(x) = λ_{I}^{−1}ϕ_{In}^{∗}ι_{r}F (ψ_{I}A_{I})(x/λ_{I}), we obtain

|ϕ_{I}^{∗} \frac{∂f_{I}^{∗}ψ_{I}A_{I}}{∂λ_{I}} |_{δ_{I_{−}}} (x) ≤ \{ \begin{matrix}4K \frac{λ_{I}^{2}|x|}{(λ_{I}^{2} + |x|^{2})^{2}} & if |x| < \frac{1}{2}N_{I}^{−1}λ_{I}^{1/2} \\ 0 & if |x| ≥ \frac{1}{2}N_{I}^{−1}λ_{I}^{1/2}, \end{matrix}

where K ≡ ∥F (ψ_{I}A_{I})∥_{L^{∞}(X_{I},g_{I})} is bounded by a constant C independent of λ_{I} by Lemma 3.9. But g = g̃_{I_{−}} on B_{I}(\frac{1}{2}N_{I}^{−1}λ_{I}^{1/2}) ⊂ X′′_{I_{−}}, and moreover, the metrics g̃_{I_{−}}, g_{I_{−}}, and

δ_{I_{−}} are equivalent over the ball B_{I}(\frac{1}{2}N_{I}^{−1}λ_{I}^{1/2}), with constants depending at most on x_{I}. Thus, we obtain the integral estimate

∫_{X′_{I_{−}}} |\frac{∂f_{I}^{∗}A′}{∂λ_{I}} |_{g}^{2} dV_{g} ≤ Cλ_{I}^{2},

and so, combining these bounds, we have ∥\frac{∂}{∂λ_{I}} Â′∥_{L^{2}(U_{2},g_{0})} ≤ Cλ_{I}, completing Step 2.

Step 3. Estimate of ∂f_{0}^{∗} ⋯ f_{I}^{∗}Â′_{I}/∂λ_{I}. We have Â′_{I} = f_{I_{+}}^{∗} ⋯ f_{J}^{∗}ψ_{J}A_{J} over V_{3} ≡ f_{I_{+}}^{−1} ∘ ⋯ ∘ f_{J}^{−1}(X′_{J}) ⊂ B′_{I_{+}} ⊂ X_{I}, with J > I. We denote Â′ = f_{0}^{∗} ⋯ f_{I}^{∗}Â′_{I} and observe that \frac{∂}{∂λ_{I}} Â′ = f_{0}^{∗} ⋯ f_{I_{−}}^{∗} \frac{∂}{∂λ_{I}} f_{I}^{∗}Â′_{I} over U_{3} ≡ f_{0}^{−1} ∘ ⋯ ∘ f_{I}^{−1}(V_{3}) ⊂ X_{0}. Thus,

\frac{∂f_{I}^{∗}Â′_{I}}{∂λ_{I}} = −λ_{I}^{−1}f_{I}^{∗}ι_{r}F (Â′_{I}) = −λ_{I}^{−1}f_{I}^{∗}ι_{r}F (f_{I_{+}}^{∗} ⋯ f_{J}^{∗}ψ_{J}A_{J})
= −λ_{I}^{−1}ι_{f_{I∗}^{−1}r}f_{I}^{∗} ⋯ f_{J}^{∗}F (ψ_{J}A_{J}).

Note that ∂Â′_{I}/∂λ_{I} is supported on f_{I}^{−1}(B′_{I_{+}}) ⊂ B′_{I}.

As r = y^{μ}∂/∂y^{μ} with respect to y = ϕ_{In}^{−1} on X_{I} \{x_{Is}}, we have f_{I∗}^{−1}r = x^{μ}∂/∂x^{μ} with respect to x = ϕ_{I}^{−1} on B′_{I}. If |y| ≤ R_{0} on B′_{I_{+}}, for some constant 0 < R_{0} < ∞ depending at most on x_{I}, then |x| ≤ R_{0}λ_{I} on f_{I}^{−1}(B′_{I_{+}}). Thus, |f_{I∗}^{−1}r|_{g_{I_{−}}} ≤ R_{0}λ_{I} on f_{I}^{−1}(B′_{I_{+}}) and so we have the pointwise bound

|\frac{∂Â′_{I}}{∂λ_{I}} |_{g_{I_{−}}} ≤ R_{0}|f_{I}^{∗} ⋯ f_{J}^{∗}F (A′_{J})|_{g_{I_{−}}}  on f_{I}^{−1}(B′_{I_{+}}).

Therefore, with the aid of repeated applications of Lemma 3.19, we find that

∥f_{0}^{∗} ⋯ f_{I_{−}}^{∗} \frac{∂Â′_{I}}{∂λ_{I}} ∥_{L^{2}(U_{3},g_{0})} ≤ C ∥\frac{∂Â′_{I}}{∂λ_{I}} ∥_{L^{2}(f_{I}^{−1}(V_{3}),g_{I_{−}})}
≤ C∥f_{I}^{∗} ⋯ f_{J}^{∗}F (A′_{J})∥_{L^{2}(f_{I}^{−1}(V_{3}),g_{I_{−}})}
≤ C∥F (A′_{J})∥_{L^{2}(X′_{J} ,g_{J})}.

and since ∥F (A′_{J})∥_{L^{2}(X′_{J} ,g_{J})} ≤ C, this gives ∥\frac{∂}{∂λ_{I}} Â′∥_{L^{2}(U_{3},g_{0})} ≤ C, completing Step 3.

Combining the results from Steps 1 to 3 then yields (a). For (b) we use the cutoff function estimate |∂ψ_{J}/∂p_{I}| ≤ Cλ_{I}^{−1/2}, J = I_{−}, I. The vector field r is replaced by p = p_{I}^{μ}∂/∂y^{μ}, with respect to the coordinates y = ϕ_{In}^{−1}. Then, f_{I∗}^{−1}p = λ_{I}p_{I}^{μ}∂/∂x^{μ} with respect to the coordinates x = ϕ_{I}^{−1} and we have the vector field estimate |f_{I∗}^{−1}p| ≤ R_{0}λ_{I} on f_{I}^{−1}(B′_{I_{+}}). The required bound then follows by an argument similar to that of (a). □

3.8. Derivatives with respect to bundle gluing parameters. The purpose of this section is to obtain estimates for the derivatives of the almost ASD connections A′ and Â′ with respect to the bundle gluing parameters ρ_{I} ∈ Gl_{I}, I > 0. These estimates may be extracted from [D-K, §7.2] and we include them here for completeness.

Since we wish to differentiate a family of connections A′(ρ_{I}) on a family of G-bundles P (ρ_{I}) with respect to the gluing parameters ρ_{I} ∈ Gl_{I}, we first pull this family back to an equivalent family on a fixed bundle, say P (ρ̄_{I}), as described in [D-K, p. 296]. Let ρ̄_{I} ∈ Gl be a given gluing parameter: then points ρ_{I} in a small neighbourhood of ρ̄_{I} in Gl_{I} can be written in the form ρ = ρ̄_{I} exp(v), where v ∈ V_{I} ≡ ad P_{I}|_{x_{Is}} ≃ 𝔤. One regards the fibres of P_{I_{−}} and P_{I} as being identified by ρ̄_{I} and so v may considered as a local section of both P_{I_{−}} and P_{I}, covariantly constant with respect to the connections A′_{I_{−}}, A′_{I}.

We digress in order to construct a set of cutoff functions {γ_{I}} on X such that ∑_{I∈ℐ} γ_{I} = 1. These cutoffs will be needed here and again in §5.1 for patching together certain integral operators over the X_{I} to give an integral operator over X. Choose a bump function γ ∈ C^{∞}(ℝ^{1}) such that γ(t) = 1 if t ≥ 2 and γ(t) = 0 if t ≤ \frac{1}{2}. Define a cut-off function γ_{λ} ∈ C^{∞}(ℝ^{4}) by

(3.52)  γ_{λ}(x) ≡ γ(|x|/λ^{1/2}),  x ∈ ℝ^{4}.

Now define C^{∞} cutoff functions γ_{I} on each summand X_{I} by setting

(3.53)  γ_{I} ≡ (ϕ_{Is}^{−1})^{∗}(1 − γ_{λ_{I}}) ∏_{I_{+}}(ϕ_{I_{+}}^{−1})^{∗}γ_{λ_{I_{+}}} on X_{I},

where the factor (ϕ_{Is}^{−1})^{∗}(1 − γ_{λ_{I}}) is omitted when I = 0. Note that γ_{I} = 0 on the balls B_{Is}(\frac{1}{2}λ_{I}^{1/2}) and B_{I+}(\frac{1}{2}λ_{I_{+}}^{1/2}) in X_{I}. We extend γ_{I} to a C^{∞} cutoff function on X_{I} by zero

on these balls and by 1 on the complement of the larger balls B_{Is}(2λ_{I}^{1/2}) and B_{I+}(2λ_{I_{+}}^{1/2}) in X_{I}; then extend by zero outside X′′_{I} ⊂ X to give γ_{I} ∈ C^{∞}(X). By construction, we have ∑_{I∈ℐ} γ_{I} = 1 on X, with a slight abuse of notation. Indeed, note that f_{I} maps the annulus Ω_{I}(\frac{1}{2}λ_{I}, 2λ_{I}) around the point x_{I} in X_{I_{−}} onto the annulus Ω_{Is}(\frac{1}{2}λ_{I}, 2λ_{I}) around the south pole x_{Is} in X_{I}. Then, f_{I}^{∗}γ_{I} + γ_{I_{−}} = 1 on each annulus Ω_{I}. Lastly, note that there is a constant C, depending at most on the metric g_{0}, such that

(3.54)  |dγ_{I}|_{g_{I}} ≤ Cλ_{I}^{−1/2} on Ω_{I}, Ω_{Is} and ∥dγ_{I}∥_{L^{p}(X_{I},g_{I})} ≤ C\overline{λ}^{2/p−1/2}.

for any 1 ≤ p < ∞. Define gauge transformations u_{I_{−}}(v) on Aut P_{I_{−}}|_{X′_{I}} and u_{I}(v) on Aut P_{I}|_{X′_{I}} by setting

(3.55)  u_{I_{−}}(v) = \{ \begin{matrix}exp(γ_{I}v) & on Ω_{I} \\ 1 & on X′_{I_{−}} \ Ω_{I} \end{matrix} ,
u_{I}(v) = \{ \begin{matrix}exp(−γ_{I_{−}}v) & on Ω_{Is} \\ 1 & on X′_{I} \ Ω_{Is}. \end{matrix}

Note that u_{I} has a natural extension to a gauge transformation of P_{I} over all of X_{I} — equal to exp(−v) on B_{Is}(N_{I}^{−1}λ_{I}^{1/2}), the ball enclosed by the annulus Ω_{Is}. Similarly for the gauge transformation u_{I_{−}}. After identifying the bundles and base manifolds over Ω = Ω_{I} = Ω_{Is}, we have u_{I_{−}}u_{I}^{−1} = exp((γ_{I_{−}} + γ_{I})v) = exp(v). Hence, relative to the flat connections A′_{I_{−}}, A′_{I}, the gauge transformations u_{I} differ by a constant bundle automorphism over Ω and so their action on the connection A′(ρ̄_{I}) is the same: u_{I_{−}}(A′(ρ̄_{I}))|_{Ω} = u_{I}(A′(ρ̄_{I}))|_{Ω}. Therefore, while the automorphisms u_{I} do not patch together to give a global automorphism of P (ρ̄_{I}), their actions on the connection A′(ρ̄_{I}) do. Indeed, we can define a connection A′(ρ̄_{I}, v) on P (ρ̄_{I}) by

(3.56)  A′(ρ̄_{I}, v) = \{ \begin{matrix}u_{I_{−}}(A′(ρ̄_{I})) & on X′_{I_{−}} \\ u_{I}(A′(ρ̄_{I})) & on X′_{I}. \end{matrix}

If ρ_{I} = ρ̄_{I} exp(v), the connections A′(ρ̄_{I}, v) and A′(ρ_{I}) are gauge equivalent [D-K, p. 296]. Thus, as desired, we have an equivalent family of connections A′(ρ̄_{I}, v) on the fixed connected sum bundle P = P (ρ̄_{I}). Let L_{I} ⊂ Gl_{I} be a coordinate neighbourhood and suppose ρ̄_{I} ∈ L_{I}. Then

(3.57)  𝔤 ⊃ B_{𝔤} ⟶ L_{I} ⊂ Gl_{I},  v ⟼ ρ_{I}(v) ≡ ρ̄_{I} exp(v)

is a coordinate chart centred at ρ̄_{I}, where B_{𝔤} is the unit ball in 𝔤, and there is a C^{∞} embedding

(3.58)  𝔤 ⊃ B_{𝔤} ⟶ A_{X,P}^{∗} ,  v ⟼ A′(ρ̄_{I}, v).

It remains to consider the derivative of the family A′(ρ̄_{I}, v) with respect to v.

Recall that if u = u(s) is a one-parameter family of gauge transformations, B is a fixed connection, and B^{u}(s) is the induced one-parameter family of gauge transformed connections, then dB^{u}/ds(0) = d_{B^{u}}(u^{−1}u̇(0)), where u^{−1}u̇(0) ∈ Ω^{0}(X, ad P). Although the u_{I_{−}}, u_{I} are not globally defined gauge transformations, this differentiation formula still applies to the one-parameter families u_{I_{−}}(s) = u_{I_{−}}(sv) and u_{I}(s) = u_{I}(sv). Therefore, we have

(3.59)  \frac{∂A′}{∂v} (ρ̄_{I}) ≡ \frac{d}{ds}A′(ρ̄_{I}, sv)|_{s=0} = \{ \begin{matrix}d_{A′}(γ_{I}v) & on X′_{I_{−}} ∩ Ω \\ −d_{A′}(γ_{I_{−}}v) & on X′_{I} ∩ Ω \\ 0 & on X \ Ω. \end{matrix}

This leads to the following estimate for the derivative of the family A′(ρ_{I}) with respect to the gluing parameters ρ_{I}; a related and more general estimate is given by Lemma 7.2.49 in [D-K].

Proposition 3.28. Let 2 ≤ p < 4 and suppose that 4 ≤ q < ∞ is determined by 1/4 + 1/q = 1/p. There is a constant c = c(g_{0}, p, 𝒯) such that

(a) c|v|λ_{I}^{2/p−1} ≤ ∥∂A′/∂v∥_{L^{q}(X,g)} ≤ c^{−1}|v|λ_{I}^{2/p−1},

(b) c|v|λ_{I}^{2/p−1/2} ≤ ∥∂A′/∂v∥_{L^{p}(X,g)} ≤ c^{−1}|v|λ_{I}^{2/p−1/2}.

Proof. Note that γ_{I_{−}} + γ_{I} = 1 on Ω and so d_{A′}(γ_{I}v) = −d_{A′}(γ_{I_{−}}v) on Ω. Moreover, d_{A′}(γ_{I}v) = dγ_{I} ∧ v on Ω, and so we have ∥d_{A′}(γ_{I}v)∥_{L^{q}(X,g)} = |v| · ∥dγ_{I}∥_{L^{q}(X,g)}. From Eq. (3.54) there is a constant c > 0 independent of λ_{I} such that

c|v|λ_{I}^{2/q−1/2} ≤ ∥\frac{∂A′}{∂v} ∥_{L^{q}(X,g)} ≤ c^{−1}|v|λ_{I}^{2/q−1/2},

since ∥∂A′/∂v∥_{L^{q}(X,g)} = ∥d_{A′}(γ_{I}v)∥_{L^{q}(X,g)}. Then (a) follows since 2/q − 1/2 = 2/p − 1, and likewise for (b). □

Using the conformal maps f_{J}, we pull back the family A′ = A′(ρ̄_{I}, v) on the fixed bundle P over X to a family Â′(ρ̄_{I}, v) on the fixed bundle P̂ over X_{0}.

Proposition 3.29. If 2 ≤ p < 4, there is a constant C = C(g_{0}, p, 𝒯) such that for any t ∈ 𝒯, ∥∂Â′/∂v∥_{L^{p}(X_{0},g_{0})} ≤ Cλ_{I}^{2/p−1/2}.

Proof. Since ∂A′/∂v = 0 outside the annulus Ω_{Is} ⊂ X′_{I}, Proposition 3.28 gives

∥\frac{∂A′}{∂v} ∥_{L^{p}(X′_{I},g_{I})} ≤ Cλ_{I}^{2/p−1/2}.

But Â′ = f_{0}^{∗} ⋯ f_{I}^{∗}A′ on U ≡ f_{0}^{−1} ∘ ⋯ ∘ f_{I}^{−1}(X′_{I}) ⊂ X_{0}, and so Lemma 3.19 gives

∥f_{0} ⋯ f_{I}^{∗} \frac{∂A′}{∂v} ∥_{L^{p}(U,g_{0})} ≤ C ∥\frac{∂A′}{∂v} ∥_{L^{p}(X′_{I},g_{I})} .

Combining these estimates gives the desired bound. □

3.9. Derivatives with respect to lower moduli. In this section we obtain L^{p} estimates for the derivatives of the connections A′, Â′, and the self-dual curvature F^{+,g}(A′) with respect to the “lower moduli parameters” t_{I}. The bundle P_{I} carrying the family of connections {A_{I}(t_{I})}_{t_{I}∈T_{A_{I}}} can be assumed to be fixed with respect to the parameters t ∈ T_{A_{I}} since the space T_{A_{I}} — an open ball in H_{A_{I}}^{1} centred at 0 — is contractible. However, the local sections σ_{I+}(t_{I}) are defined by the connections A_{I}(t_{I}) (together a choice of point in P_{I}|_{x_{I_{+}}} ) and will vary with t_{I}. Thus, the bundle gluing maps for the connected sum bundle P, defined by σ_{I_{+}}(t_{I}) ↦ σ_{I_{+}s}ρ̃_{I_{+}}(t_{I}) (suppressing the identification map f_{I} : Ω_{I_{+}} → Ω_{I_{+}s}), will in general vary with t_{I}. We may suppose that the remaining parameters are fixed and thus we obtain a family of connections A′(t_{I}) on a family of bundles P (t_{I}). The difficulty, of course, is that unless we have a family of connections defined on a fixed bundle, we cannot define the derivative ∂A′/∂t_{I}. Problems such as these are discussed in [D87, p. 423]. For our purposes, we note the bundles are all isomorphic and as T_{A_{I}} is contractible, the connections A′(t_{I}) could be pulled back by bundle isomorphisms h_{I} ∈ Hom(P (0), P (t_{I})) to an equivalent family h_{I}^{∗}A′(t_{I}) on the fixed bundle P (0) and then we could define

(3.60)  \frac{∂A′}{∂t_{I}} ≡ \frac{∂h_{I}^{∗}A′}{∂t_{I}} .

Since any two such families h_{I}(t_{I}) of bundle isomorphisms would differ by a family of automorphisms of the fixed bundle P (0), then ∂A′/∂t_{I} would give a well-defined tangent vector to B_{P (0)}^{∗} at [A′(t_{I})] using (3.60). Naturally, the analogous remarks apply to the family of connections Â′(t_{I}) on the bundles P̂(t_{I}).

In our case, a family of isomorphisms h_{J}(t_{I}) : P_{J}(0) → P_{J}(t_{I}) may be described quite explicitly, in a manner similar to that of §3.8, and these will give a gauge equivalent family of connections h_{I}^{∗}A′(t_{I}), ĥ_{I}^{∗}Â′(t_{I}) on fixed bundles P (0), P̂(0) respectively, although just as in §3.8, the isomorphisms h_{J}(t_{I}) will not patch together to give a global isomorphism of P (0) with P (t_{I}) or P̂(0) with P̂(t_{I}). Nonetheless Eq. (3.60) still makes sense and this allows us to estimate the length of the tangent vector ∂A′/∂t_{I} in terms of derivatives of the local connection one-forms, as desired. Let h_{I}(t_{I}) : P_{I}(0) → P_{I}(t_{I}) be a family of bundle isomorphisms represented locally by σ_{I_{+}}(0) ↦ σ_{I_{+}}(t)θ_{I_{+}}(t_{I}). Then h_{I}^{∗}A_{I}(t_{I}) is an equivalent family on the fixed bundle P_{I}(0), with

σ_{I_{+}}(0)^{∗}h_{I}(t_{I})^{∗}A_{I}(t_{I}) = θ_{I_{+}}(t_{I})^{−1}σ_{I_{+}}(t_{I})^{∗}A_{I}(t_{I})θ_{I_{+}}(t_{I}) + θ_{I_{+}}(t_{I})^{−1}dθ_{I_{+}}(t_{I}).

Note that while the local connection one-forms σ_{I_{+}}(t_{I})^{∗}A_{I}(t_{I}) are in radial gauge, this will not in general be the case for the one-forms σ_{I_{+}}(0)^{∗}h_{I}(t_{I})^{∗}A_{I}(t_{I}). We next consider the variation in the bundle gluing maps ρ̃_{I_{+}}(t_{I}) induced by the variation in σ_{I_{+}}(t_{I}) with t_{I}. Over X_{I}, we replace θ_{I_{+}}(t_{I}) above by θ_{I_{+}}(t_{I}) exp(γ_{I_{+}}v_{I}(t_{I})) and over X_{I_{+s}} define h_{I}(t_{I}) by right multiplication with exp(γ_{I}v_{I}(t_{I})). Recalling the notation of §3.8, v_{I} : T_{A_{I}} → 𝔤 is a smooth map with v(0) = 0 defined (for small enough T_{A_{I}}) by the identity ρ_{I_{+}}(t_{I}) = ρ_{I_{+}}(0) exp(v_{I}(t_{I})). Lastly, for J ≠ I, I_{+}, we set h_{J}(t_{I}) = 1. Then, for the remainder of this article, we require that the derivatives ∂A′/∂t_{I} be defined by (3.60).

This understood, we obtain the following estimates for the derivatives with respect to the parameters t_{I} of the connections A′ and Â′ and for the g-self-dual curvature F^{+,g}(A′). The proofs are straightforward, following the pattern in §3.7, and so are omitted.

Proposition 3.30. Let 1 ≤ p < ∞. For sufficiently small b_{0}, there exists a constant C = C(g_{0}, p, 𝒯) such that for any t ∈ 𝒯,

(a) ∥∂A′/∂t_{I} − ∂A_{I}/∂t_{I}∥_{L^{p}(X′′_{I},g_{I})} ≤ Cλ_{I}^{2/p}

(b) ∥∂A′/∂t_{I}∥_{L^{p}(X,g)} ≤ C.

Proposition 3.31. Let 1 ≤ p < ∞. For sufficiently small b_{0}, there is a constant C = C(g_{0}, p, 𝒯) such that for any t ∈ 𝒯, ∥∂F^{+,g}(A′)/∂t_{I}∥_{L^{p}(X,g)} ≤ Cλ̄^{2/p−1/2}.

Proposition 3.32. For sufficiently small b_{0}, there is a constant C = C(g_{0}, N, 𝒯) such that for any t ∈ 𝒯, ∥∂Â′/∂t_{I}∥_{L^{2}(X_{0},g_{0})} ≤ C.

Proof. Let U ≡ f_{0}^{−1} ⋯ f_{I}^{−1}(X′_{I}) ⊂ X_{0} and note note that \frac{∂}{∂t_{I}} Â′ = \frac{∂}{∂t_{I}} f_{0}^{∗} ⋯ f_{I}^{∗}ψ_{I}A_{I}, which is \frac{∂}{∂t_{I}} f_{0}^{∗} ⋯ f_{I}^{∗}ψ_{I}A_{I} = f_{0}^{∗} ⋯ f_{I}^{∗} \frac{∂}{∂t_{I}} ψ_{I}A_{I} on U and zero elsewhere. Now

∥f_{0}^{∗} ⋯ f_{I}^{∗} \frac{∂ψ_{I}A_{I}}{∂t_{I}} ∥_{L^{2}(U,g_{0})} ≤ C ∥\frac{∂ψ_{I}A_{I}}{∂t_{I}} ∥_{L^{2}(X′_{I},g_{I})}

by Lemma 3.19 and so the result follows. □

3.10. Differentials of the approximate gluing maps. We close this Chapter by summarising the results of the preceding sections and record our bounds for the differentials of the approximate gluing maps 𝒥′ (which follow by combining Propositions 3.25, 3.30, and 3.28)) and 𝒥̂′ (which follow by combining Propositions 3.27, 3.32, and 3.29).

Theorem 3.33. Let 𝒥′ : 𝒯 → B_{X,k}^{∗} be the approximate gluing map t ↦ [A′(t)]. Assume

b_{I} = 4N_{I}λ_{I}^{1/2} for all I. Then for sufficiently small λ_{0} and any t ∈ 𝒯, there is a constant C = C(g_{0}, 𝒯) such that the following estimates hold.

(a) ∥D𝒥′(∂/∂t_{I}^{α})∥_{L^{2}(X,g)} ≤ C,

(b) ∥D𝒥′(∂/∂ρ_{I}^{β})∥_{L^{2}(X,g)} ≤ Cλ_{I}^{1/2},

(c) ∥D𝒥′(∂/∂x_{I}^{μ})∥_{L^{2}(X,g)} ≤ C,

(d) ∥D𝒥′(∂/∂λ_{I})∥_{L^{2}(X,g)} ≤ C.

Theorem 3.34. Let 𝒥̂′ : 𝒯 → B_{X_{0},k}^{∗} be the approximate gluing map t ↦ [Â′(t)]. Let

b_{I} = 4N_{I}λ_{I}^{1/2} for all I. Then for sufficiently small λ_{0} and any t ∈ 𝒯, there is a constant C = C(g_{0}, 𝒯) such that the following estimates hold.

(a) ∥D𝒥̂′(∂/∂t_{I}^{α})∥_{L^{2}(X_{0},g_{0})} ≤ C,

(b) ∥D𝒥̂′(∂/∂ρ_{I}^{β})∥_{L^{2}(X_{0},g_{0})} ≤ Cλ_{I}^{1/2},

(c) ∥D𝒥̂′(∂/∂x_{I}^{μ})∥_{L^{2}(X_{0},g_{0})} ≤ C,

(d) ∥D𝒥̂′(∂/∂λ_{I})∥_{L^{2}(X_{0},g_{0})} ≤ C.

4. Bubble Tree Compactification of the Anti-self-dual Moduli Space

In order to describe the ends of the moduli space M_{X_{0},k}(g_{0}) one customarily appeals to the Uhlenbeck compactification \overline{M}_{X_{0},k}^{u}(g_{0}). This allows one to give quite explicit descriptions of the parts of the ends away from the diagonals in the symmetric products M_{X_{0},k}(g_{0}) × s^{l}(X_{0}) appearing in the compactification, as for example in [D86, §V] and [D-K, §8.2]. These examples consider ideal boundary points of the form (A_{0}, x_{1}, …, x_{l}), where the x_{i} are distinct points with multiplicity 1 and A_{0} is a g_{0}-ASD connection over X_{0}. Open neighbourhoods of (A_{0}, x_{1}, …, x_{l}) in \overline{M}_{X_{0},k}^{u}(g_{0}) are then constructed by gluing standard one-instantons onto A_{0}.

In order to construct open neighbourhoods of ideal boundary points corresponding to the diagonals of \overline{M}_{X_{0},k}^{u}(g_{0}) we must employ the iterated gluing construction of Chapters 3 and 5. This strategy is mentioned briefly in [D-K, §8.2]. The construction gives a homeomorphism 𝒥̂ : 𝒯^{0}/Γ → \overline{𝒱}, where \overline{𝒱} is an open neighbourhood of a boundary point in \overline{M}_{X_{0},k}^{u}(g_{0}) — a ‘gluing neighbourhood’. In order to use this procedure to describe the ends of \overline{M}_{X_{0},k}^{u}(g_{0}), we need to show that \overline{M}_{X_{0},k}^{u}(g_{0}) is covered by finitely many such gluing neighbourhoods. In particular, we need to show that any point in M_{X_{0},k}(g_{0}) which is sufficiently close to the ideal boundary (with respect to the Uhlenbeck topology) lies in the image of a gluing map 𝒥̂. This is accomplished in two steps: Step 1. We show that any sequence {A_{α}} of g_{0}-ASD connections over X_{0} converging weakly to a limit (A_{0}, x_{1}, …, x_{m_{0}}) determines a sequence of metrics {g_{α}} and a sequence {Ǎ_{α}} of g_{α}-ASD connections over a connected sum X ≡ #_{I∈ℐ}X′_{Iα} which converges strongly to a limit (A_{I})_{I∈ℐ}, in the sense of [D-K, §7.3]. Here, (X, g_{α}) is conformally equivalent to (X_{0}, g_{0}), for all α, and is defined exactly as in §3.3 and §3.5. Step 2. We apply an analogue of Theorem 7.3.2 [D-K] to show that the new sequence {Ǎ_{α}} is D_{q}-convergent, q ≥ 4, in the sense of [D-K, §7.3]. The appropriate analogue of Theorem 7.2.62 [D-K] then shows that the points [A_{α}] ∈ M_{X,k}(g_{α}) lie in the image of some 𝒥 for sufficiently large α. Consequently, the points [A_{α}] ∈ M_{X_{0},k}(g_{0}) lie in the image of the corresponding map 𝒥̂, for some parameter space 𝒯^{0}/Γ. The choice of parameter space 𝒯^{0}/Γ is essentially determined by (A_{I})_{I∈ℐ}, which we call the strong or bubble tree limit of the sequence {A_{α}}.

In this Chapter we discuss Step 1 and describe the bubble tree compactification of the moduli space of anti-self-dual SU(2) connections — the extension to the general case of compact, semi-simple Lie groups being straightforward. Step 2 is discussed in §§5.1 and 5.2 after the necessary analytical framework has been established. Throughout this Chapter, we suppose only that X_{0} is a closed, oriented, simply-connected C^{∞} four-manifold, g_{0} is a C^{∞} metric, and G = SU(2).

4.1. Uhlenbeck compactification. We recall the definition of the Uhlenbeck compactification [D-K] and describe some of the related convergence results we will need for our description of the bubble tree compactification.

Definition 4.1. An Uhlenbeck ideal g_{0}-ASD connection on a G bundle P over X_{0} with c_{2}(P) = k ≥ 0 is a pair (A_{0}, Z_{0}), where A_{0} is a g_{0}-ASD connection on a G bundle P_{0} over X_{0} with c_{2}(P_{0}) = k_{0} ≥ 0 and Z_{0} = {x_{i}}_{i=1}^{m_{0}} is a (possibly empty) set of points in X_{0} with multiplicities k_{i} ≥ 1, for i = 1, …, m_{0}, such that ∑_{i=0}^{m_{0}} k_{i} = k. The curvature density of (A_{0}, Z_{0}) is defined to be the Borel measure

(4.1)  μ(A_{0}, Z_{0}) = |F (A_{0})|_{g_{0}}^{2} + 8π^{2}δ_{Z_{0}},

where δ_{Z_{0}} ≡ ∑_{i=1}^{m_{0}} k_{i}δ_{x_{i}}, so that the total mass of μ(A_{0}, Z_{0}) is 8π^{2}k. Setting l = k_{1} + ⋯ + k_{m} and repeating points according to their multiplicity, one obtains an element (x_{1}, …, x_{l}) of the symmetric product s^{l}(X_{0}).

Definition 4.2. Let {A_{α}}_{α=1}^{∞}, be a sequence of g_{0}-ASD connections on a G bundle P over X_{0} with c_{2}(P) = k ≥ 0 and let (A_{0}, Z_{0}) be an ideal g_{0}-ASD connection on P. Then the sequence {A_{α}} converges weakly to (A_{0}, Z_{0}) if:

(a) The sequence {μ_{α}}_{α=1}^{∞} converges to μ(A_{0}, Z_{0}) in the weak-* topology on measures;

(b) There is a sequence of C^{∞} bundle maps γ_{α} : P_{0}|_{X_{0}\Z_{0}} → P |_{X_{0}\Z_{0}} such that γ_{α}^{∗}A_{α} converges in C^{∞} on compact subsets of X_{0} \ Z_{0} to the connection A_{0}. Equivalently, require that for any integer n ≥ 1, there is a sequence of L_{n+1}^{2} bundle maps γ_{α} such that γ_{α}^{∗}A_{α} converges in L_{n,loc}^{2} on X_{0} \ Z_{0} to A_{0}.

Via the natural extension of Definition 4.2 to sequences of ideal connections, the set of all Uhlenbeck ideal g_{0}-ASD connections of fixed second Chern class k, IM_{X_{0},k}(g_{0}) ≡ ∐_{l=0}^{k}(M_{X_{0},k−l}(g_{0}) × s^{l}(X_{0})), is endowed with a metrisable topology. Let \overline{M}_{X_{0},k}^{u}(g_{0}) be the closure of M_{X_{0},k}(g_{0}) in IM_{X_{0},k}(g_{0}). According to [D-K, Theorem 4.4.4], any infinite sequence in M_{X_{0},k}(g_{0}) has a weakly convergent subsequence with limit point in \overline{M}_{X_{0},k}^{u}(g_{0}), and in particular, the latter space is compact [D-K, Theorem 4.4.3].

For our description of the bubble tree compactification, we will need the following minor extension of the convergence result in Theorem 4.4.4 [D-K] and its cousin, Proposition 9.4.2 [D-K], which allows for a sequence of metrics {g_{α}} converging to g_{0} in C^{∞}. The proof employs standard arguments well described in [D-K, §4.4] and is left to the reader.

Proposition 4.3. Let {U_{α}}_{α=1}^{∞} be an exhaustion of the punctured manifold X_{0} \ {p} by an increasing sequence {U_{α}}_{α=1}^{∞} of precompact open sets, so that U_{1} ⋐ U_{2} ⋐ ⋯ ⊂ X_{0} \{p} and ∪_{α=1}^{∞}U_{α} = X_{0}\{p}. Let {g_{α}}_{α=1}^{∞} be a sequence of metrics on the subsets U_{α} converging in C^{r} (r ≥ 3) on compact subsets of X_{0} \{p} to a C^{r} metric g_{0} on X_{0}. Let P be a G bundle over X_{0} \ {p} and let {A_{α}}_{α=1}^{∞} be a sequence of g_{α}-ASD connections on the restrictions P |_{U_{α}}. If there is a constant M < ∞ such that

∫_{U_{α}} |F (A_{α})|_{g_{α}}^{2} dV_{g_{α}} ≤ M for all α,

then there is a set of points Z_{0} = {x_{i}}_{i=1}^{m_{0}} ⊂ X_{0} and a g_{0}-ASD connection A_{0} on a G bundle P_{0} over X_{0} such that a subsequence {A_{α}}_{α=1}^{∞} converges weakly to (A_{0}, Z_{0}). □

The mass of the Uhlenbeck limit (A_{0}, Z_{0}) in Proposition 4.3 is 8π^{2} times an integer and may be computed from the weakly convergent sequence {A_{α}}_{α=1}^{∞} by

(4.2)  lim_{n→∞} lim_{α→∞} ∫_{V_{n}} |F (A_{α})|_{g_{α}}^{2} dV_{g_{α}},

where {V_{n}}_{n=1}^{∞} is any exhaustion of X_{0} \{p} by an increasing sequence of precompact open subsets.

4.2. Conformal blow-ups. Given a sequence of g_{0}-anti-self-dual connections on a G bundle P over X_{0} with curvature densities concentrating near a set of ‘singular points’ in X_{0}, we define associated sequences of mass centres and scales. In a manner analogous to Chapter 3, we then obtain sequences of ‘conformal blow-up maps’ f_{Iα} (defined exactly as in §3.3) which resolve these singularities in a sense that will be made precise below and in §4.3. As will become evident, the process of applying conformal blow-ups may need to be iterated before the singularities are completely ‘resolved’.

Let us commence by defining the first level conformal blow-ups. Suppose {A_{α}}_{α=1}^{∞} is a sequence of g_{0}-anti-self-dual connections over X_{0} with weak limit (A_{0}, Z_{0}). Let us consider the behaviour of the sequence {A_{α}}_{α=1}^{∞} in M_{X_{0},k}(g_{0}) near the singular set Z_{0} = {x_{i}}_{i=1}^{m_{0}} in more detail. If the point x_{i} has multiplicity k_{i}, then

(4.3)  lim_{r→∞} lim_{α→∞} ∫_{B(x_{i},r)} |F (A_{α})|_{g_{0}}^{2} dV_{g_{0}} = 8π^{2}k_{i}.

Choose constants d_{0}, r_{0} such that

(4.4)  0 < d_{0} ≤ min_{i≠j} dist_{g_{0}}(x_{i}, x_{j}) and 0 < r_{0} < \frac{1}{4} min\{1, ϱ_{0}, d_{0}\}.

We next define mass centres and scales of g_{0}-anti-self-dual connections restricted to the fixed ball B(x_{i}, r_{0}) ⊂ X_{0} by appropriately modifying the previous definitions of mass centres and scales of §3.2 for g_{1}-anti-self-dual connections over 𝕊^{4}. First, note that

(4.5)  lim_{α→∞} ∫_{B(x_{i},r_{0})} (|F (A_{α})|_{g_{0}}^{2} − |F (A_{0})|_{g_{0}}^{2}) dV_{g_{0}} = 8π^{2}k_{i}.

Choose a frame v_{i} in F X_{0}|_{x_{i}} and let q = ϕ_{x_{i}}^{−1} be the associated geodesic normal coordinate chart. For each i, define a sequence of mass centres {x_{iα}}_{α=1}^{∞} in B(x_{i}, r_{0}) by x_{iα} ≡ ϕ_{x_{i}}(q_{iα}), where q_{iα} = Centre [A_{α}|_{B(x_{i},r_{0})}] ∈ ℝ^{4} and

(4.6)  Centre [A_{α}|_{B(x_{i},r_{0})}] ≡ \frac{1}{8π^{2}k_{i}} ∫_{B(x_{i},r_{0})} q (|F (A_{α})|_{g_{0}}^{2} − |F (A_{0})|_{g_{0}}^{2}) dV_{g_{0}}.

Define a sequence of scales {λ_{iα}}_{α=1}^{∞} in (0, ∞) by setting λ_{iα} = Scale[A_{α}|_{B(x_{i},r_{0})}], where

(4.7) Scale^{2}[A_{α}|_{B(x_{i},r_{0})}] ≡ \frac{1}{8π^{2}k_{i}} ∫_{B(x_{i},r_{0})} |q − q_{iα}|^{2} (|F (A_{α})|_{g_{0}}^{2} − |F (A_{0})|_{g_{0}}^{2}) dV_{g_{0}}.

As in §3.2., Eq. (4.7) leads to a Tchebychev inequality:

(4.8)  ∫_{B(x_{i},r_{0})\B(x_{iα},Rλ_{iα})} (|F (A_{α})|_{g_{0}}^{2} − |F (A_{0})|_{g_{0}}^{2}) dV_{g_{0}} ≤ 8π^{2}k_{i}R^{−2},  R ≥ 1.

Hence, if R ≫ 1 and α is sufficiently large, the balls B(x_{iα}, Rλ_{iα}) contain most of the 8π^{2}k_{i} quantity of A_{α}-energy bubbling off at x_{iα}.

Remark 4.4. Other choices of scale function are possible. For example, we might have chosen λ_{iα} to be the radius of the ball centred at x_{iα} containing A_{α}-energy 8π^{2}(k_{i} − \frac{1}{2}). As in [D83], a cutoff function is required in order to regularise this definition.

Thus, we obtain a sequence of scales {λ_{iα}}_{α=1}^{∞} associated to the sequences of mass centres {x_{iα}}_{α=1}^{∞} and connections {A_{α}}_{α=1}^{∞}. Moreover, Eq. (4.3) implies that the sequence x_{iα} converges to x_{i} and that the sequence of scales λ_{iα} converges to zero. Choose a sequence of frames v_{iα} ∈ F X_{0}|_{x_{iα}} converging to the frame v_{i} ∈ F X_{0}|_{x_{i}} and let ϕ_{x_{iα}}^{−1} be the corresponding geodesic normal coordinate charts. Let f_{x_{iα}} ≡ ϕ_{in} ∘ c_{λ_{iα}} ∘ ϕ_{x_{iα}}^{−1}, where c_{λ_{iα}} is the dilation of ℝ^{4} given by x ↦ x/λ_{iα}, let g̃_{iα} be the approximately round metric on X′_{iα} defined as in §3.5, let P_{iα} = (f_{x_{iα}}^{−1})^{∗}P be the induced G bundle over X′_{iα}, and let A_{iα} = (f_{x_{iα}}^{−1})^{∗}A_{α} be the induced g̃_{iα}-anti-self-dual connection on P_{iα}. We call the maps f_{x_{iα}} conformal blow-ups.

We obtain a sequence of open subsets X′_{iα} which exhaust X_{i} \ {x_{is}}, a sequence of metrics {g̃_{iα}}_{α=1}^{∞}, and a sequence of g̃_{iα}-anti-self-dual connections {A_{iα}}_{α=1}^{∞} over the X′_{iα}. The sequence {g̃_{iα}}_{α=1}^{∞} converges in C^{∞} on compact subsets of X_{i} \ {x_{is}} to the standard round metric g_{i} on X_{i} ≡ 𝕊^{4}. Let {g_{α}}_{α=1}^{∞} be the sequence of C^{∞} metrics, defined as in §3.5, on the connected sum X ≡ #_{i=0}^{m_{0}}X′_{iα}, defined as in §3.3, and let {Ǎ_{α}}_{α=1}^{∞} be the induced sequence of g_{α}-anti-self-dual connections over X. We call the connected sums (X, g_{α}) conformal blow-ups of (X_{0}, g_{0}).

There is a uniform upper bound on the L^{2} norms ∥F (A_{iα})∥_{L^{2}(X′_{iα},g̃_{iα})} since

(4.9)  ∫_{X′_{iα}} |F (A_{iα})|2 ; g̃_{iα} dVg̃_{iα} = ∫_{B(x_{iα},Nλ_{iα}^{1/2})} |F (A_{α})|_{g_{0}}^{2} dV_{g_{0}} ≤ 8π^{2}(k_{i} + 1/2),

for sufficiently large α by Eq. (4.3), while Eqs. (4.5) and (4.8) give a lower bound

(4.10)  ∫_{X′_{iα}} |F (A_{iα})|2 ; g̃_{iα} dVg̃_{iα} = ∫_{B(x_{iα},Nλ_{iα}^{1/2})} |F (A_{α})|_{g_{0}}^{2} dV_{g_{0}} ≥ 8π^{2}(k_{i} − 1/2).

Proposition 4.3 provides a subsequence {A_{iα}}_{α=1}^{∞} which converges weakly to an ideal g_{i}- anti-self-dual connection (A_{i}, Z_{i}) over X_{i}, where Z_{i} = {x_{ij}}_{j=1}^{m_{i}}. The energy bound of Eq. (4.8) ensures that Z_{i} ⊂ X_{i} \ {x_{is}}. Let μ_{i} = μ(A_{i}, Z_{i}) be the associated singular measure on X_{i} and note that its mass may be computed by

∫_{X_{i}} dμ_{i} = lim_{R→∞} lim_{α→∞} ∫_{B(x_{in},R)} |F (A_{iα})|2 ; g̃_{iα} dVg̃_{iα}.

Since this must be 8π^{2} times an integer, Eqs. (4.9) and (4.10) imply that μ_{i} has mass 8π^{2}k_{i}, where k_{i} = ∑_{j=0}^{m_{i}} k_{ij}, A_{i} is a g_{i}-anti-self-dual connection on a bundle P_{i} over X_{i} with c_{2}(P_{i}) = k_{i0}, and each point x_{ij} has multiplicity k_{ij}.

Remark 4.5. It is not strictly necessary that we construct a sequence of honest metrics g_{α} over the connected sums X = #_{i=0}^{m_{0}}X′_{iα} above; a sequence of conformal structures [g_{α}] constructed as in §3.5 would suffice and this would eliminate the need for the choice of conformal factors over the necks. In any case, the actual limits obtained are independent of such choices.

The above conformal blow-up construction produces a sequence of g̃_{x_{iα}}-anti-self-dual connections A_{x_{iα}} on increasing subsets X′_{iα} of the four-sphere X_{i} with weak g_{i}-anti-self-dual limit (A_{i}, Z_{i}). With the inverse process of gluing in mind, we describe a modified choice of conformal blow-ups which yield centred limits (Ã_{i}, Z̃_{i}). First, a technical lemma concerning the variation of geodesic normal coordinate charts with their coordinate centres is required. The proof uses Taylor’s theorem and is left to the reader.

Lemma 4.6. Let X_{0} be a closed C^{∞} n-manifold with metric g_{0} and injectivity radius ϱ_{0}. Let x_{0} ∈ X, let v_{0} ∈ F X|_{x_{0}}, and let x = exp_{v_{0}}^{−1} be the geodesic normal coordinate chart on B(x_{0}, ϱ_{0}) defined by the frame v_{0}. Suppose x_{1} ∈ B(x_{0}, ϱ_{0}/4) and p = exp_{v_{0}}^{−1}(x_{1}), so that dist_{g_{0}}(x_{1}, x_{0}) = |p|. We now define two coordinate charts on B(x_{1}, ϱ_{0}/2):

(a) Let v_{1} ∈ F X|_{x_{1}} be the frame obtained by parallel translating v_{0} along the geodesic joining x_{0} to x_{1}, and let w = exp_{v_{1}}^{−1} on B(x_{1}, ϱ_{0}/2);

(b) Let τ_{p} be the translation on ℝ^{n} given by q ↦ q − p, and let w̄ = τ_{p} ∘ exp_{v_{0}}^{−1} on B(x_{1}, ϱ_{0}/2). Then the coordinates w̄ converge to w in C^{∞} on B(x_{0}, ϱ_{0}/4) as p → 0: |w̄^{μ} −w^{μ}| = O(|w||p|), |∂w̄^{μ}/∂w^{α} −δ_{α}^{μ}| = O(p), and for all m ≥ 2, ∂^{m}w̄^{μ}/∂w^{α_{1}} ⋯ ∂w^{α_{m}} = O(p).

Next, we define the mass centre and scale of a positive Borel measure μ on ℝ^{4} by

(4.11)  p = Centre [μ] ≡ ∫_{ℝ^{4}} x dμ and λ^{2} = Scale^{2}[μ] ≡ ∫_{ℝ^{4}} |x − p|^{2} dμ.

Let Θ be the product connection over X_{i}. The proof of the following lemma describes how to choose conformal blow-ups which produce centred limits.

Lemma 4.7. Let {A_{α}} be a sequence of g_{0}-anti-self-dual connections over X_{0} with weak limit (A_{0}, Z_{0}), where Z_{0} = {x_{i}}_{i=1}^{m_{0}} is non-empty. Choose r_{0} as in Eq. (4.4). Then for each x_{i} ∈ Z_{0}, the sequence {A_{α}} determines a sequence of points {w_{iα}} converging to x_{i}, a sequence of frames v_{iα} ∈ F X_{0}|_{w_{iα}} converging to a frame v_{i} ∈ F X_{0}|_{x_{i}}, and a sequence of scales {κ_{iα}} converging to zero such that the following holds. Fix N > 4, let f_{w_{iα}} be the corresponding sequences of conformal blow-ups, and let A_{w_{iα}} be the induced sequence of g̃_{w_{iα}}-anti-self-dual connections with weak g_{i}-anti-self-dual limit (Ã_{i}, Z̃_{i}) over the four-sphere X_{i}. The limit (Ã_{i}, Z̃_{i}) has the following properties:

(a) If Ã_{i} ≠ Θ, then Ã_{i} is centred;

(b) If Ã_{i} = Θ, then the corresponding singular measure μ̃_{i} is centred.

Proof. (a) We begin by defining, exactly as before, a sequence of points {x_{iα}} converging to x_{i}, a sequence of frames v_{iα} ∈ F X_{0}|_{x_{iα}} converging to a frame v_{i} ∈ F X_{0}|_{x_{i}}, and a sequence of scales {λ_{iα}} converging to zero. Let f_{x_{iα}} be the corresponding sequences of conformal blow-ups and let A_{x_{iα}} be the induced sequence of g̃_{x_{iα}}-anti-self-dual connections with weak g_{i}-anti-self-dual limit (A_{i}, Z_{i}) over X_{i}. Suppose Centre [A_{i}] = p_{i} and Scale[A_{i}] = ν_{i}.

Case 1. Z_{i} = ∅. Recall that f_{x_{iα}} = ϕ_{in} ∘ c_{λ_{iα}} ∘ ϕ_{x_{iα}}^{−1}, A_{x_{iα}} = (f_{x_{iα}}^{−1})^{∗}A_{α}, and g̃_{x_{iα}} = λ_{iα}^{−2}(f_{x_{iα}}^{−1})^{∗}g_{0}. Define h_{i} = ϕ_{in} ∘ c_{ν_{i}} ∘ τ_{p_{i}} ∘ ϕ_{in}^{−1} and set f̄_{w_{iα}} = h_{i} ∘ f_{x_{iα}}. Then

f̄_{w_{iα}} = ϕ_{in} ∘ c_{λ_{iα}ν_{i}} ∘ τ_{p_{i}λ_{iα}} ∘ ϕ_{x_{iα}}^{−1} = ϕ_{in} ∘ c_{κ_{iα}} ∘ ϕ̄_{w_{iα}}^{−1},

where w_{iα} ≡ ϕ_{x_{iα}}(p_{i}λ_{iα}), κ_{iα} ≡ λ_{iα}ν_{i}, and ϕ̄_{w_{iα}} ≡ ϕ_{x_{iα}} ∘ τ_{p_{i}λ_{iα}}^{−1}. Thus, f̄_{w_{iα}} provides a

diffeomorphism from the small ball B(w_{iα}, Nκ_{iα}^{1/2}) in X_{0} to the open subset B(x_{in}, Nκ_{iα}^{−1/2}) of X_{i}. The sequence of points {w_{iα}} converges to x_{i} and the sequence of scales {κ_{iα}} converges to zero. As in §3.5, define a sequence of metrics on the increasing subsets B(x_{in}, Nκ_{iα}^{−1/2}) by ḡ_{w_{iα}} ≡ κ_{iα}^{−2}h_{1}^{2}(f̄_{w_{iα}}^{−1})^{∗}g_{0}. Then ḡ_{w_{iα}} converges to the standard metric g_{i} in C^{∞} on compact subsets of X_{i}\{x_{is}}. Define a sequence of ḡ_{w_{iα}}-anti-self-dual connections over the balls B(x_{in}, Nκ_{iα}^{−1/2}) by Ā_{w_{iα}} ≡ (f̄_{w_{iα}}^{−1})^{∗}A_{α}, and observe that Ā_{w_{iα}} = (h_{i}^{−1})^{∗}A_{x_{iα}}. The sequence {Ā_{w_{iα}}} converges to the centred connection (h_{i}^{−1})^{∗}A_{i} in C^{∞} on compact subsets of X_{i} \ {x_{is}}.

It remains to replace the chart w̄ ≡ ϕ̄_{w_{iα}}^{−1} on B(w_{iα}, ϱ_{0}/2) by a geodesic normal coordinate chart w ≡ ϕ_{w_{iα}}^{−1}. Choose a frame v′_{iα} ∈ F X_{0}|_{w_{iα}} by parallel translating the frame v_{iα} ∈ F X_{0}|_{x_{iα}} along the geodesic connecting x_{iα} and w_{iα}, noting that dist_{g_{0}}(x_{iα}, w_{iα}) = |p_{i}|λ_{iα}. Thus, as α → ∞, the coordinate chart w̄ converges in C^{∞} on B(x_{i}, ϱ_{0}/4) to the geodesic normal coordinate chart w in the sense of Lemma 4.6. Define a new sequence of conformal blow-up maps by setting f_{w_{iα}} = ϕ_{in} ∘ c_{κ_{iα}} ∘ ϕ_{w_{iα}}^{−1}, and define corresponding sequences of connections and metrics on the balls B(x_{in}, Nκ_{iα}^{−1/2}) by A_{w_{iα}} = (f_{w_{iα}}^{−1})^{∗}A_{α} and g̃_{w_{iα}} = κ_{iα}^{−2}h_{1}^{2}(f_{w_{iα}}^{−1})^{∗}g_{0}. Lemma 4.6 implies that the sequences {g̃_{w_{iα}}} and {A_{w_{iα}}} converge in C^{∞} on compact subsets of X_{i} \ {x_{is}} to the metric g_{i} and centred g_{i}-anti-self-dual connection Ã_{i} ≡ (h_{i}^{−1})^{∗}A_{i}. This completes the proof of (a) in Case 1. Case 2. Z_{i} ≠ ∅. The proof is similar to that of Case 1. Let Z̃_{i} = h_{i}^{−1}(Z_{i}). Then the sequences {Ā_{w_{iα}}} and {A_{w_{iα}}} converge in C^{∞} on compact subsets of X_{i} \ (Z̃_{i} ∪ {x_{is}}) to the centred connection (h_{i}^{−1})^{∗}A_{i}.

(b) One sets Centre [μ_{i}] = p_{i}, Scale[μ_{i}] = ν_{i}, and essentially repeats the proof of Part (a) for the sequence of measures μ_{x_{iα}} ≡ |F (A_{x_{iα}})|_{g_{0}}^{2}. □

Remark 4.8. In the sequel, we require that the conformal blow-up maps be chosen as in Lemma 4.7. However, to conserve notation, we will relabel the points w_{iα} and scales κ_{iα} by x_{iα} and λ_{iα}, respectively, and the limit (Ã_{i}, Z̃_{i}) by (A_{i}, Z_{i}).

A technical point that we have not addressed above is that, just as in [P-W], the weak limit of the sequence {A_{iα}} apparently depends on certain choices of parameters in the conformal blow-up construction:

(1) Neck width parameter N. This was only included in this Chapter for the sake of consistency with the gluing construction of Chapters 3 and 5: we could just as well have set N = 2, say.

(2) Radius r_{0}. Following [P-W], the dependency is removed by letting r_{0} → 0. The conformal blow-up process gives a sequence of points {x_{iα}(r_{0})}, scales {λ_{iα}(r_{0})}, blow-up maps {f_{x_{iα}}(r_{0})}, metrics {g̃_{iα}(r_{0})}, and connections {A_{iα}(r_{0})}. The sequence of connections {A_{iα}(r_{0})} converges to an ideal g_{i}-anti-self-dual limit (A_{i}(r_{0}), Z_{i}(r_{0})), for any fixed r_{0} > 0. We now let r_{0} → 0 and by a standard diagonal argument, we obtain a weakly convergent subsequence {A_{iα}(r_{0})} with weak limit (A_{i}, Z_{i}), say.

(3) Frames v_{iα} and v_{i}. The construction is SO(4) equivariant: Rotating the frames v_{iα} ∈ F X|_{x_{iα}} and v_{i} ∈ F X_{x_{i}} by elements of SO(4) induces an SO(4) action on the connections A_{iα} and A_{i} as described in §3.2.

There is one final issue which will be important in our later discussion of alternative modes of convergence for sequences of anti-self-dual connections: we must exclude the possibilty that curvature is lost over the necks Ω_{i} arising in the conformal blow-up process described above. Of course, the curvature can only bubble off with masses equal to an integer multiple of 8π^{2}, so it suffices to show that we can choose the neck parameters to ensure that the curvature masses over the necks are strictly less than 8π^{2}. So, consider again the sequence {A_{α}}_{α=1}^{∞} of g_{0}-anti-self-dual connections over X_{0} with weak limit (A_{0}, Z_{0}), where Z_{0} = {x_{i}}_{i=1}^{m_{0}}, and let {A_{iα}}_{α=1}^{∞} be the corresponding sequences of g̃_{iα}-anti-self-dual connections over X′_{iα} having weak limits (A_{i}, Z_{i}), where Z_{i} = {x_{ij}}_{j=1}^{m_{j}}. Let {λ_{iα}}_{α=1}^{∞} be the sequence of scales associated to the sequence of connections {A_{α}}_{α=1}^{∞} and the singular point x_{i} ∈ Z_{0}. Given this set-up, standard arguments yield the following curvature estimates near x_{i}:

Lemma 4.9. Given ε > 0, there exist positive constants R_{0}, r_{1}, and α_{0} with the following significance. For large enough R_{0}, small enough r_{1} and large enough α_{0}, then R_{0}λ_{iα} < r_{1} for any α ≥ α_{0} and the following holds.

(a) |∥F (A_{iα})∥_{L^{2}(B(x_{i},R_{0}),g̃_{iα})}^{2} − 8π^{2}k_{i}| < ε^{2},

(b) |∥F (A_{α})∥_{L^{2}(B(x_{i},R_{0}λ_{iα}),g_{0})}^{2} − 8π^{2}k_{i}| < ε^{2},

(c) ∥F (A_{iα})∥_{L^{2}(Ω(x_{i},R_{0},r_{1}λ_{iα}^{−1}),g̃_{iα})} < ε,

(d) ∥F (A_{α})∥_{L^{2}(Ω(x_{i},R_{0}λ_{iα},r_{1}),g_{0})} < ε.

Thus, we have the following curvature estimate which ensures that in the limit there is no ‘curvature loss’ over the necks Ω_{i}. (In particular, if A_{iα} converges weakly to (A_{i}, Z_{i}), then the singular set Z_{i} ⊂ X_{i} does not contain the south pole x_{is}.)

Corollary 4.10. Given ε > 0 and N > 4, there is an α_{0} > 0 with the following significance. If Ω_{iα} ≡ Ω(x_{iα}, N^{−1}λ_{iα}^{1/2}, Nλ_{iα}^{1/2}) and, B′_{iα} ≡ B(x_{iα}, Nλ_{iα}^{1/2}), then for any α ≥ α_{0}, we have

(a) ∥F (A_{α})∥_{L^{2}(Ω_{iα},g_{0})} < ε, and

(b) |∥F (A_{α})∥_{L^{2}(B′_{iα},g_{0})}^{2} − 8π^{2}k_{i}| < ε.

Lastly, we note that the conformal blow-up process may of course be iterated if the singular sets Z_{i} are non-empty. In the next section we show that after repeating the conformal blow-up process at most k times, we obtain a sequence of g_{α}-anti-self-dual connections {Ǎ_{α}} which is strongly convergent. Indeed, given the weakly convergent sequence {A_{iα}}_{α=1}^{∞} over the X′_{iα} near a point x_{ij} with multiplicity k_{ij} in the singular set Z_{i} ⊂ X_{i}, the second-level process differs from the first-level only in minor technical details: We define sequences of centres x_{ijα} = ϕ_{x_{ij}}(q_{ijα}) converging to x_{ij} and scales λ_{ijα} converging to zero, now using the metrics g̃_{iα} and a coordinate chart ϕ_{x_{ij}} on X_{i} given by ϕ_{x_{ij}} = ϕ_{in} ∘τ_{q_{ij}}^{−1} where ϕ_{in}(q_{ij}) = x_{ij}. The blow-up maps are then defined using coordinate charts on X_{i} given by ϕ_{x_{ijα}} = ϕ_{in} ∘τ_{q_{ijα}}^{−1} and setting f_{x_{ijα}} = ϕ_{ijn} ∘c_{λ_{ijα}} ∘ϕ_{x_{ijα}}^{−1}. We then proceed exactly as before and similarly for all higher-level blow-ups.

4.3. Bubble tree compactification. By analogy with the arguments of [T88, §5] and [P-W], we define a bubble tree compactification for the moduli space M_{X_{0},k}(g_{0}) of antiself-dual connections. First, we need an appropriate notion of an “ideal connection”:

Definition 4.11. A bubble tree ideal g_{0}-anti-self-dual connection A of second Chern class k over X_{0} is determined by the following data.

(a) An oriented tree ℐ with a finite set of vertices {I}, including a base vertex 0, and a set of edges {(I_{−}, I)}. Each vertex I is labelled with an integer k_{I} ≥ 0 such that (i) ∑_{I∈ℐ} k_{I} = k, (ii) If I > 0 is a terminal vertex, then k_{I} > 0, (iii) There are at most k terminal vertices, excluding the base vertex.

(b) A (2m − 1)-tuple (A_{I}, x_{I})_{I∈ℐ}, where m is the number of vertices in ℐ.

(c) If I = 0, then A_{0} is a g_{0}-anti-self-dual connection on a G bundle P_{0} over X_{0} with c_{2}(P_{0}) = k_{0} ≥ 0.

(d) If I > 0, then

(i) A_{I} is either the product connection Θ or a centred g_{I}-anti-self-dual connection on a G bundle P_{I} over the sphere X_{I} ≡ 𝕊^{4} with c_{2}(P_{I}) = k_{I}, where g_{I} is the standard round metric,

(ii) x_{I} is a point in X_{0} if I_{−} = 0 and a point in X_{I_{−}} \ {x_{Is}} if I_{−} > 0.

(e) If I > 0 and A_{I} = Θ, then there are at least 2 outgoing edges emanating from that vertex.

Definition 4.11 should be compared with the construction of approximately anti-self-dual connections in §3.3. The ideal connection (A_{I}, x_{I})_{I∈ℐ} is often written as (A_{I})_{I∈ℐ}. Heuristically, we may view an ideal g_{0}-anti-self-dual connection A = (A_{I})_{I∈ℐ} as a ‘connection’ over the join ∨_{I∈ℐ}X_{I}, where each sphere X_{I} is attached to the lower level X_{I_{−}} by identifying the south pole x_{Is} with the point x_{I} ∈ X_{I_{−}}. Let Z_{I_{−}} ⊂ X_{I_{−}} denote the set of ‘attachment points’ x_{I} in X_{0}, if I_{−} = 0, or points x_{I} in X_{I_{−}} \ {x_{Is}}, if I_{−} > 0. Let m_{I} be the number of points in Z_{I}, i.e., the number of outgoing edges emanating from vertex I.

Second, we need an appropriate notion of convergence. Let X ≡ #_{I∈ℐ}X_{I} be the connected sum defined in §3.3 by a set of scales \{λ_{Iα}\}_{I∈ℐ}, with \overline{λ}_{α} → 0 as α → ∞, and a fixed neck parameter N. Similarly, if {g_{α}} is the corresponding sequence of C^{∞} metrics on X defined in §3.5, then g_{α} converges to g_{I} in C^{∞} on compact subsets of X_{I} \ (Z_{I} ∪ {x_{Is}}) for each I ≥ 0. Following [D-K, §7.3.1], we consider the following modes of convergence for sequences of anti-self-dual connections over X.

Definition 4.12. Let {A_{α}}_{α=1}^{∞} be a sequence of g_{α}-anti-self-dual connections on a fixed bundle P with c_{2}(P) = k over the connected sum X = #_{I∈ℐ}X_{I}.

(a) If Y ∈ s^{k}(X) is a multiset in ∪_{I∈ℐ}X_{I} \ (Z_{I} ∪ {x_{Is}}), the sequence {A_{α}} converges weakly to ((A_{I}, x_{I})_{I∈ℐ}, Y) if the gauge equivalence classes [A_{α}] converge in C^{∞} to ([A_{I}])_{I∈ℐ} over compact subsets of ∪_{I∈ℐ}X_{I} \ (Z_{I} ∪ {x_{Is}} ∪ Y) and if the curvature densities converge,

|F (A_{α})|_{g_{α}}^{2} ⟶ ∑_{I∈ℐ}|F (A_{I})|_{g_{I}}^{2} + 8π^{2}δ_{Y} ,

over compact subsets of ∪_{I∈ℐ}X_{I} \ (Z_{I} ∪ {x_{Is}}).

(b) The sequence {A_{α}} converges strongly to the limit (A_{I}, x_{I})_{I∈ℐ} if it converges weakly to (A_{I}, x_{I})_{I∈ℐ} (with no singular set Y) and if ∑_{I∈ℐ} c_{2}(P_{I}) = c_{2}(P). Here, the A_{I} are g_{I}-anti-self-dual connections on G bundles P_{I} over X_{I} with c_{2}(P_{I}) = k_{I}.

We let BM_{X_{0},k}(g_{0}) denote the set bubble tree ideal g_{0}-anti-self-dual connection over X_{0} of total second Chern class k. Thus, each point of BM_{X_{0},k}(g_{0}) is represented by a (2m − 1)-tuple (A_{I}, x_{I})_{I∈ℐ}, with m being the total number of vertices of the tree ℐ.

Definition 4.13. We say that a sequence {A_{α}}_{α=1}^{∞} of g_{0}-anti-self-dual connections on a G bundle P over X_{0} with c_{2}(P) = k converges strongly to a bubble tree ideal g_{0}-anti-self-dual connection (x_{I}, A_{I})_{I∈ℐ} in BM_{X_{0},k}(g_{0}) if there exist sequences of conformal blow-ups {f_{Iα}}_{I∈ℐ} with the following property. Let {g_{α}} be the induced sequence of C^{∞} metrics in the conformal class [g_{0}] on the connected sum X = #_{I∈ℐ}X_{I}. Let {Ǎ_{α}} denote the induced sequence of g_{α}-anti-self-dual connections over X. Then we require that the sequence of metrics {g_{α}} converges in C^{∞} on compact sets of X_{I} \(Z_{I} ∪{x_{Is}}) to the metric g_{I}, I ≥ 0, and that the sequence of connections {Ǎ_{α}} converges strongly to the ideal g_{0}-anti-self-dual connection (A_{I}, x_{I})_{I∈ℐ}.

This definition of convergence extends to the space of bubble tree ideal connections BM_{X_{0},k}(g_{0}), which is then endowed with a second countable Haussdorf topology. Define the bubble tree compactification \overline{M}_{X_{0},k}^{τ}(g_{0}) to be the closure of M_{X_{0},k}(g_{0}) in BM_{X_{0},k}(g_{0}).

Theorem 4.14. The space \overline{M}_{X_{0},k}^{τ}(g_{0}) is compact.

The result follows from the special case below.

Theorem 4.15. Any infinite sequence in M_{X_{0},k}(g_{0}) has a strongly convergent subsequence with limit point in \overline{M}_{X_{0},k}^{τ}(g_{0}).

Proof. The argument is similar to the proof of Proposition 5.3 in [T88]. Fix a G bundle P over X_{0} with c_{2}(P) = k > 0 and let {A_{α}}_{α=1}^{∞} be a sequence of g_{0}-anti-self-dual connections on P. The main point is to repeatedly apply conformal blow-ups f_{Iα} until we obtain a sequence of induced metrics g_{α} over a connected sum X, with (X, g_{α}) conformally equivalent to (X_{0}, g_{0}), and a sequence of induced g_{α}-anti-self-dual connections over X, denoted by {Ǎ_{α}}, which is strongly convergent. We adopt the convention below that subsequences are immediately relabelled.

Step 1. There is a subsequence {A_{α}} which converges weakly to an ideal g_{0}-anti-self-dual connection (A_{0}, Z_{0}), with Z_{0} = {x_{i}}_{i=1}^{m_{0}} corresponding to a point in the symmetric product s^{k}(X_{0}). If Z_{0} = ∅ then we are done, so assume that m_{0} ≥ 1. Let k_{i} be the multiplicity of x_{i} and note that 0 < k_{i} ≤ k. For each i and large enough α, the connection A_{α} determines a set of mass centres {x_{iα}}_{i=1}^{m_{0}}, with x_{iα} → x_{i}, and a set of scales {λ_{iα}}_{i=1}^{m_{0}}, with λ_{iα} → 0 as α → ∞. Fix a neck width parameter N > 4, choose a sequence of frames v_{iα} ∈ F X_{0}|_{x_{iα}} converging to a frame v_{i} ∈ F X_{0}|_{x_{i}}, and let {f_{iα}}_{i=1}^{m_{0}} be the conformal blow-up maps defined by these centres, frames, scales, and parameter N. If X = #_{i=0}^{m_{0}}X′_{iα}, then (X, g_{α}) is the conformal blow-up of (X_{0}, g_{0}) determined by the maps f_{iα}. Let P now denote the induced G bundle over X, let Ǎ_{α} denote the induced g_{α}-anti-self-dual connection over X, and let A_{iα} be the restriction of Ǎ_{α} to the open subset X′_{iα}.

The sequence [A_{iα}] has a weakly convergent subsequence, again denoted [A_{iα}], with weak limit (A_{i}, Z_{i}), where Z_{i} corresponds to a point in s^{k_{i}}(X_{i}). Corollary 4.10 implies that no mass is lost over the neck Ω_{i}. Hence, if each Z_{i} = ∅, i > 0, then we have ∑_{i=0}^{m_{i}} k_{i} = k, the sequence [A_{iα}] converges strongly to [A_{i}], and we proceed to the Final Step. Otherwise, Z_{i} ≠ ∅ for some i and we proceed to Step 2. Step 2. For some i, Step 1 produces a non-empty singular set Z_{i} = {x_{ij}}_{j=1}^{m_{i}}. Let k_{ij} be the multiplicity of the point x_{ij}, let c_{2}(A_{i}) = k_{i0}, and note that ∑_{j=0}^{m_{i}} k_{ij} = k_{i} > 0. Let μ_{i} be the singular measure associated with (A_{i}, Z_{i}). We now consider two cases, depending on whether or not A_{i} is the flat product connection Θ over X_{i}. Case (a) A_{i} = Θ. Since Scale[μ_{i}] = 1, the diameter of the set Z_{i} must be positive and so this case can only occur if m_{i} > 1. Let k_{ij} be the multiplicity of the point x_{ij} and note that as m_{i} > 1 we must have max_{j} k_{ij} ≤ k − 1. Case (b) A_{i} ≠ Θ. Therefore, k_{i0} = c_{2}(A_{i}) > 0 and so we again must have max_{j} k_{ij} ≤ k − 1, since ∑_{j=0}^{m_{i}} k_{ij} = k_{i} ≤ k.

For large enough α, the connection A_{iα} determines a set of mass centres {x_{ijα}}_{i=1}^{m_{i}}, with x_{ijα} → x_{ij}, and a set of scales {λ_{ijα}}_{i=1}^{m_{i}}, with λ_{ijα} → 0 as α → ∞. Let {f_{ijα}}_{j=1}^{m_{i}} be the conformal blow-up maps defined by these centres, scales, and parameter N. Let P denote the induced G bundle over the new connected sum X = #_{i=0}^{m_{0}}X′_{iα}#_{j=1}^{m_{i}}X′_{ijα}, let Ǎ_{α} denote the induced g_{α}-anti-self-dual connection over X, and let {A_{ijα}} be the induced sequence of g_{α}-anti-self-dual connections over the open subsets X′_{ijα} of the spheres X_{ij}.

The sequence [A_{ijα}] has a weakly convergent subsequence with weak limit (A_{ij}, Z_{ij}), with no loss of mass over the necks Ω_{ijα}. If each Z_{ij} = ∅, j = 1, …, m_{i}, then we have ∑_{j=0}^{m_{i}} k_{ij} = k_{i}, the sequence [A_{ijα}] converges strongly to [A_{ij}], and the blow-up process terminates at the vertices A_{ij}. Otherwise, Z_{ij} ≠ ∅ for some j and we proceed to Step 3. Step l: 3 ≤ l ≤ k. For some multi-index I of length |I| = l − 1, Step l − 1 produces a non-empty singular set Z_{I} = {x_{Ij}}_{j=1}^{m_{I}} contained in the sphere X_{I}. The sequence [A_{Iα}] has a weak limit (A_{I}, Z_{I}), where Z_{I} corresponds to a point in s^{k_{I}} (X_{I}). Let k_{Ij} be the multiplicity of the point x_{Ij}, let c_{2}(A_{I}) = k_{I0}, and note that ∑_{j=0}^{m_{I}} k_{Ij} = k_{I} > 0. Let μ_{I} be the singular measure associated with (A_{I}, Z_{I}).

Case (a) A_{I} = Θ. Since Scale[μ_{I}] = 1, the diameter of the set Z_{I} must be positive. Hence, m_{I} > 1 and so we have

(4.12)  max_{j} k_{Ij} ≤ k − l + 1,  |Ij| = l, 1 ≤ l ≤ k.

Case (b) A_{I} ≠ Θ. Therefore, k_{I0} = c_{2}(A_{I}) > 0 and so Eq. (4.12) again holds, since ∑_{j=0}^{m_{I}} k_{Ij} = k_{I} ≤ k.

Eq. (4.12) implies that the conformal blow-up process terminates completely after at most k steps.

For large enough α, the connection A_{Iα} determines a set of mass centres {x_{Ijα}}_{j=1}^{m_{I}} in X_{I} \ {x_{Is}}, with x_{Ijα} → x_{Ij}, and a set of scales {λ_{Ijα}}_{j=1}^{m_{I}}, with λ_{Ijα} → 0 as α → ∞. Let {f_{Ijα}}_{j=1}^{m_{I}} be the conformal blow-up maps defined by these centres, scales, and parameter N. Let P denote the induced G bundle over the connected sum X = #_{I}X′_{Iα}#_{j=1}^{m_{i}}X′_{Ijα}, let Ǎ_{α} denote the induced g_{α}-anti-self-dual connection over X, and let {A_{Ijα}} be the induced sequence of g_{α}-anti-self-dual connections over the open subsets X′_{Ijα} of the spheres X_{Ij}.

The sequence [A_{Ijα}] has a weakly convergent subsequence with weak limit (A_{Ij}, Z_{Ij}), with no loss of mass over the necks Ω_{Ijα}. If each Z_{Ij} = ∅, j = 1, …, m_{I}, then we have ∑_{j=0}^{m_{I}} k_{Ij} = k_{I}, the sequence [A_{Ijα}] converges strongly to [A_{Ij}], the blow-up process terminates at the vertices A_{Ij}, and we proceed to the Final Step. Otherwise, proceed to Step l + 1. Final Step. After performing at most k conformal blow-ups, we obtain a sequence of g_{α}-anti-self-dual connections {Ǎ_{α}} over a connected sum X = #_{I∈ℐ}X′_{Iα}. The sequence {Ǎ_{α}} converges strongly to a bubble tree limit (A_{I}, x_{I})_{I∈ℐ}, since the singular points have all been blown up and there has been no mass loss over the necks Ω_{Iα}. □

Plainly, the compactification \overline{M}_{X_{0},k}^{τ}(g_{0}) is “larger” than the Uhlenbeck compactification \overline{M}_{X_{0},k}^{u}(g_{0}). Indeed, there is an obvious surjective map

(4.13)  π : \overline{M}_{X_{0},k}^{τ}(g_{0}) ⟶ \overline{M}_{X_{0},k}^{u}(g_{0})

obtained by sending a bubble tree ideal connection (A_{I}, x_{I})_{I∈ℐ} to the corresponding Uhlenbeck ideal connection (A_{0}, x_{1}, …, x_{m_{0}}). The multiplicity of x_{i} ∈ X_{0} is the sum of the second Chern classes of the anti-self-dual connections A_{I} attached to the subtree lying above the vertex i.

Corollary 4.16. The map π : \overline{M}_{X_{0},k}^{τ}(g_{0}) → \overline{M}_{X_{0},k}^{u}(g_{0}) is continuous.

4.4. D_{q} convergence and strong convergence. We will need one further notion of convergence in order to show that every point of the moduli space M_{X,k}(g) lies in the image of the gluing map 𝒥 constructed in Chapter 5. Let P be a G bundle over a closed manifold X with metric g. Following [D-K, §7.2.4], fix 4 ≤ q < ∞ and let D_{q} be the metric on the space B_{X,P} given by

(4.14)  D_{q}([A], [B]) = inf_{u∈G} ∥A − u(B)∥_{L_{q}(X,g)}.

We recall the following definition of Donaldson and Kronheimer.

Definition 4.17. [D-K, p. 308] Let {λ_{Iα}}_{α=1}^{∞}, for each I > 0, be sequences of scales satisfying \overline{λ}_{α} → 0, where \overline{λ}_{α} = max_{I} λ_{Iα}, and let \{A_{α}\}_{α=1}^{∞} be a sequence of connections on a fixed G bundle P → X, where X ≡ #_{I∈ℐ}X′_{I} and X_{I} = 𝕊^{4} if I > 0. The connected sum X has a sequence of metrics {g_{α}}_{α=1}^{∞} defined by the sequence of scales {λ_{Iα}}_{α=1}^{∞}, a sequence of points {x_{Iα}}_{α=1}^{∞}, where the x_{Iα} converge with respect to the fixed metric g_{I} to a point x_{I} ∈ X_{I_{−}}, and a neck width parameter N. Assume that the connections A_{α} are g_{α}-ASD with respect to the sequence of metrics {g_{α}}_{α=1}^{∞} on X. Then the sequence {A_{α}}_{α=1}^{∞} is D_{q}-convergent to (A_{I}, x_{I})_{I∈ℐ} if D_{q}([A_{α}|_{X′′_{I}}], [A_{I}|_{X′′_{I}}]) → 0 as α → ∞.

D_{q} convergence is called “L^{q} convergence” in [D-K]. The result below explains the relationship between strong convergence and D_{q} convergence.

Theorem 4.18. [D-K, p. 309] Let {A_{α}}_{α=1}^{∞} be a sequence of connections on a bundle P → X which are ASD with respect to the sequence of metrics {g_{α}}_{α=1}^{∞} determined by the sequences of scales \{λ_{Iα}\}, where \overline{λ}_{α} → 0. Then the sequence \{A_{α}\}_{α=1}^{∞} is strongly convergent if and only if it is D_{q}-convergent. □

5. Differentials of the gluing maps

In this Chapter we obtain L^{2} estimates for the differentials of the gluing maps 𝒥̂ : 𝒯/Γ → M_{X_{0},k}^{∗}. These give C^{0} bounds for the components of the L^{2} metric g on the bubbling ends of M_{X_{0},k}^{∗}(g_{0}) and allow us to complete the proofs of Theorems 1.1 and 1.2. In particular, for the remainder of the article, the hypotheses of Theorem 1.1 are assumed to be in effect.

5.1. Construction of the gluing maps. In this section we construct the gluing maps 𝒥 : 𝒯/Γ → M_{X,k}^{∗}(g) and 𝒥̂ : 𝒯/Γ → M_{X_{0},k}^{∗}(g_{0}), and set up the analytical framework required for the later sections. Our first task is to construct a right inverse to the linear operator d_{A′}^{+,g} and so we choose suitable Sobolev spaces L^{q}, L_{1}^{p} and for the remainder of this Chapter, fix

(5.1)  2 ≤ p < 4 and 4 ≤ q < ∞ so that 1/4 + 1/q = 1/p.

By hypothesis, H_{A_{I}}^{2} = 0 for all I and thus the operators d_{A_{I}}^{+,g_{I}} have right inverses P_{I}. More explicitly, if Δ_{A_{I}}^{+,g_{I}} is the Laplacian d_{A_{I}}^{+,g_{I}} (d_{A_{I}}^{+,g_{I}})^{∗} and G_{A_{I}}^{+,g_{I}} is the corresponding Green’s operator, we may set P_{I} = (d_{A_{I}}^{+,g_{I}})^{∗}G_{A_{I}}^{+,g_{I}}. A standard application of the Calderon-Zygmund theory and the Sobolev inequalities gives the following bounds.

Lemma 5.1. Assume H_{A_{I}}^{2} = 0. Then the operators P_{I} : L^{p} → L_{1}^{p} and P_{I} : L^{p} → L^{q} are bounded and there are constants C_{i} = C_{i}(A_{I}, g_{I}, p), i = 1, 2, such that

∥P_{I}ξ∥_{L^{q}(X_{I},A_{I},g_{I})} ≤ C_{1}∥P_{I}ξ∥_{L_{1}^{p}(X_{I},g_{I})} ≤ C_{2}∥ξ∥_{L^{p}(X_{I},g_{I})},  ξ ∈ L^{p}Ω^{2}(X_{I}, ad P_{I}).

We next define the C^{∞} cut-off functions to be used in the construction of a right parametrix Q for d_{A′}^{+,g} by patching together the operators P_{I} over X.

Lemma 5.2. [Lemma 7.2.10, D-K], [D-S, p. 221] For any λ > 0 and N > 4, there exists a C^{∞} function β_{λ,N} on ℝ^{4} and a constant K independent of λ, N, such that β_{λ,N}(x) = 1 if |x| ≥ \frac{1}{2}λ^{1/2} and β_{λ,N}(x) = 0 if |x| ≤ N^{−1}λ^{1/2}, and ∥dβ_{λ,N}∥_{L^{4}(ℝ^{4},δ)} ≤ K(log N)^{−3/4}.

Define C^{∞} cut-off functions β_{I} on each X_{I} by setting

(5.2)  β_{I} ≡ (ϕ_{Is}^{−1})^{∗}β_{λ_{I},N} ∏_{I_{+}}(ϕ_{I_{+}}^{−1})^{∗}β_{λ_{I_{+}} ,N} on X_{I},

where the factor (ϕ_{Is}^{−1})^{∗}β_{λ_{I},N} is omitted when I = 0. Here, the cut-off functions comprising β_{I} have been extended so that β_{I} = 1 on the complement in X_{I} of the balls B_{Is}(\frac{1}{2}λ_{I}^{1/2}) and B_{I+}(\frac{1}{2}λ_{I_{+}}^{1/2}). Also, β_{I} = 0 on the balls B_{Is}(N^{−1}λ_{I}^{1/2}) and B_{I+}(N^{−1}λ_{I_{+}}^{1/2}) in X_{I}: thus, we may extend β_{I} by zero to give β_{I} ∈ C^{∞}(X). The L^{4} estimate of Lemma 5.2 implies that

(5.3)  ∥dβ_{I}∥_{L^{4}(X_{I},g_{I})} ≤ cK(log N)^{−3/4}.

for some c = c(g_{0}, k). For the cut-off functions {γ_{I}} defined by Eqs. (3.52) and (3.53), we recall that ∑_{I} γ_{I} = 1 on X. Note also that β_{I} = 1 on the support of γ_{I}.

Define operators Q_{I} : L_{1}^{p}Ω^{+,g_{I}} (X_{I}, ad P_{I}) → L^{p}Ω^{1}(X_{I}, ad P_{I}) by setting Q_{I} = β_{I}P_{I}γ_{I} Define a right parametrix Q : ℒ_{1}^{p}Ω^{+,g}(X, ad P) → L^{p}Ω^{1}(X, ad P) for the operator d_{A′}^{+,g} by Q = ∑_{I} Q_{I}. The error operator R : L^{p}Ω^{+,g}(X, ad P) → L^{p}Ω^{+,g}(X, ad P) is then given by

(5.4)  d_{A′}^{+,g}Q = 1 + R.

Lemmas 3.15 and 5.1 then yield the following estimates for the operators Q_{I} and Q.

Lemma 5.3. There are constants C_{i} = C_{i}(g_{0}, p, 𝒯), i = 1, 2, such that for any t ∈ 𝒯 and any ξ ∈ L^{p}Ω^{+,g_{I}} (X_{I}, ad P_{I}), for (a), or any ξ ∈ L^{p}Ω^{+,g}(X, ad P), for (b), the following bounds hold.

(a) ∥Q_{I}ξ∥_{L^{q}(X_{I},g_{I})} ≤ C_{1}∥Q_{I}ξ∥_{L_{1}^{p}(X_{I},A_{I},g_{I})} ≤ C_{2}∥ξ∥_{L^{p}(X_{I},g_{I})},

(b) ∥Qξ∥_{L^{q}(X,g)} ≤ C_{1}∥Qξ∥_{ℒ_{1}^{p}(X)} ≤ C_{2}∥ξ∥_{L^{p}(X,g)}.

Next, there is an analogue of Lemma 7.2.14 [D-K] (see also [D-K, p. 294]), giving an L^{p} bound for the operator R. The proof follows easily from Lemmas 3.9, 3.12, and 5.1, and Eq. (5.3). In [D-K] it is assumed that the metrics g_{I} are flat in small neighbourhoods of the points x_{I}, but this restriction is easily removed using Lemma 3.12.

Lemma 5.4. There is a constant ε = ε(\overline{b}, N, p), with ε → 0 as N → ∞ and \overline{b} → 0 such that for any t ∈ 𝒯 and ξ ∈ L^{p}Ω^{+,g}(X, ad P), ∥Rξ∥_{L^{p}(X,g)} ≤ ε∥ξ∥_{L^{p}(X,g)}.

Thus, for the remainder of this article choose N_{0} > 4 large enough and b_{0} ≤ 1 small enough so that ε(\overline{b}, N, p) ≤ 2/3 for all \overline{b} ≤ b_{0} and N ≥ N_{0}, and fix N = N_{0} and b_{I} = 4Nλ_{I}^{1/2} for all I ∈ ℐ. We now construct a right inverse P for d_{A′}^{+,g}. Lemma 5.4 gives the (L^{p}, L^{p}) operator norm bounds ∥R∥ ≤ 2/3 and ∥(1 + R)^{−1}∥ ≤ 3. Since Q_{I} = β_{I}P_{I}γ_{I}, we have the (L^{p}, L^{q}) operator norm bound ∥Q_{I}∥ ≤ C_{I}, say, giving the (L^{p}, L^{q}) operator norm bound ∥Q∥ ≤ C ≡ ∑_{I} C_{I}. In summary, there is the following version of Proposition 7.2.35 [D-K].

Proposition 5.5. There are constants N_{0} and b_{0} such that for any N ≥ N_{0}, \overline{b} ≤ b_{0}, and t ∈ 𝒯, the operator P ≡ Q(1 + R)^{−1} : ℒ_{1}^{p}Ω^{+,g}(X, ad P) → L^{p}Ω^{1}(X, ad P) is a right inverse to d_{A′}^{+,g} and there are constants C_{i} = C_{i}(g_{0}, p, 𝒯), i = 1, 2 such that for any ξ ∈ L^{p}Ω^{+,g}(X, ad P),

∥P ξ∥_{L^{q}(X,g)} ≤ C_{1}∥P ξ∥_{ℒ_{1}^{p}(X)} ≤ C_{2}∥ξ∥_{L^{p}(X,g)}.

We next construct families of solutions to the full non-linear anti-self-dual equation over connected sums. For each t ∈ 𝒯 we seek a solution A(t) = A′(t)+a(t) to F^{+,g}(A′+a) = 0, or equivalently

(5.5)  d_{A′}^{+,g}a + (a ∧ a)^{+,g} = −F^{+,g}(A′),

where a ∈ Ω^{1}(X, ad P). If a = P ξ, with ξ(t) ∈ Ω^{+,g}(X, ad P), then this equation becomes

(5.6)  ξ + (P ξ ∧ P ξ)^{+,g} = −F^{+,g}(A′).

With the aid of Lemma 7.2.23 [D-K, p. 290] (an application of the Contraction Mapping Theorem to Eq. (5.6)) and Proposition 5.5, one easily obtains the version below of Theorem 7.2.24 [D-K].

Theorem 5.6. For sufficiently small λ_{0} < 1, sufficiently large N_{0} > 4, and sufficiently small T_{A_{I}}, I ∈ ℐ, the following holds. For any t ∈ 𝒯, there exists an L_{1}^{p} g-anti-self-dual connection A(t) = A′(t) + a(t) over X, with a(t) = P ξ(t). There are positive constants C_{i} = C_{i}(g_{0}, p, 𝒯), i = 1, 2, 3, such that

∥a∥_{L^{q}(X,g)} ≤ C_{1}∥ξ∥_{L^{p}(X,g)} ≤ C_{2}∥F^{+,g}(A′)∥_{L^{p}(X,g)} ≤ C_{3}\overline{b}^{4/p}.

We pull back the g-anti-self-dual connections A on P → X via the conformal maps f_{I} to give g_{0}-anti-self-dual connections Â = Â′ + â on P̂ → X_{0}, where Â is defined by

(5.7)  Â = f_{0}^{∗} ⋯ f_{I}^{∗}A over f_{0}^{−1} ⋯ f_{I}^{−1}(X′_{I}),

and similarly for Â′ and â. In particular, Â = Â′ + â is a solution to the g_{0}-anti-self-dual equation F^{+,g_{0}}(Â′ + â) = 0 over X_{0}, or explicitly

(5.8)  d+,g ; Â′ â + (â ∧ â)^{+,g_{0}} = −F^{+,g_{0}}(Â′),

where â ∈ Ω^{1}(X_{0}, ad P̂). Standard arguments show that the anti-self-dual connections A and Â are actually C^{∞} and that they are smooth points of the moduli spaces M_{X,k}(g) and M_{X_{0},k}(g_{0}) [D-K]:

Lemma 5.7. Let A be the g-anti-self-dual connection over X produced by Theorem 5.6 and let Â be the corresponding g_{0}-anti-self-dual connection over X_{0}. Then the following hold:

(a) The connections A and Â are C^{∞},

(b) H_{A}^{0} = 0 and H0 ; Â = 0, for small enough b_{0} and large enough N_{0},

(c) H_{A}^{2} = 0 and H2 ; Â = 0.

From §4.4, we recall that D_{q} is the metric B_{X,k} given by D_{q}([A], [B]) = inf_{u∈G} ∥A − u(B)∥_{L_{q}(X,g)}. In particular, we have the following version of Theorem 7.2.62 [D-K] (compare also Theorem 4.53 [D86]).

Theorem 5.8. Let A_{I} be g_{I}-anti-self-dual connections on G bundles P_{I} over manifolds X_{I}, I ∈ ℐ. If I = 0, then X_{0} is a closed, oriented, C^{∞} four-manifold with generic C^{∞} metric g_{0} and negative definite intersection form. If I > 0, then X_{I} = 𝕊^{4} with standard round metric g_{1} of radius 1. Let X = #_{I∈ℐ}X_{I}, the connected sum four-manifold with C^{∞} metric g (conformally equivalent to g_{0}) determined by the choice of points {x_{I}}, frames {v_{I}}, scales {λ_{I}}, and neck width parameter N. Let P be the connected sum bundle over X, where c_{2}(P) = k ≥ 1. Let \overline{λ} = max_{I∈ℐ} λ_{I}. Let T_{A_{I}} be open balls centred at 0 ∈ H_{A_{I}}^{1} , I ∈ ℐ, let Γ = ∏_{I∈ℐ} Γ_{A_{I}} and T = T_{A_{0}} × ∏_{I∈ℐ}(T_{A_{I}} × Gl_{x_{I}}), as in Eqs. (3.25) and (3.26). Then, for sufficiently small λ_{0} < 1, sufficiently large N_{0} > 4, and sufficiently small T_{A_{I}}, I ∈ ℐ, the following holds. There is a C^{∞} homeomorphism onto an open subset:

𝒥 : T/Γ ⟶ U ⊂ M_{X,P}^{∗} (g), t ⟼ [A(t)],

where A(t) = A′(t)+a(t), a(t) = P ξ(t), and ξ(t) are as in Theorem 5.6. For any ν > 0 and 4 ≤ q < ∞, the manifold T and constant λ_{0}(ν) can be chosen so that, for all \overline{λ} < λ_{0}(ν), U = {[A] ∈ M_{X,P}^{∗} (g) : D_{q}([A|_{X′′_{I}}], [A_{I}]) < ν}, for all I ∈ ℐ.

Proof. This is a straightforward generalisation of Theorem 7.2.62 [D-K] to the case of multiple connected sums (see [D-K, §7.2.8]) and a restriction to the case where G = SU(2) and b^{+}(X_{0}) = 0. The metric g_{0} is not required to be flat in small neighourhoods of the gluing sites x_{I} ∈ X_{0}. Lemma 5.7 implies that the image of 𝒥 lies in the dense open subset M_{X,P}^{∗} (g) ⊂ M_{X,P} (g). The fact that 𝒥 is C^{∞} is a calculation of the type that appears many times in §§5.3, 5.4, and 5.5. See also Appendix A [T84b] and Remark 4.24 [D86]. □

We refer to 𝒥 as a gluing map over the connected sum and its image U ⊂ M_{X,k}^{∗}(g) as a gluing neighbourhood. Moreover, 𝒥 extends to a C^{∞} gluing map on the larger parameter spaces 𝒯 and 𝒯^{0} of Eqs. (3.27) and (3.30). Further properties of these maps are described in the next section. Lastly, for the original metric g_{0} on the base four-manifold X_{0}, Theorem 5.8 takes the following form.

Corollary 5.9. Given the hypotheses of Theorem 5.8, there is a homeomorphism onto an open subset

𝒥̂ : T/Γ ⟶ V ⊂ M_{X_{0},P̂}^{∗}(g_{0}), t ⟼ [Â(t)],

where V ⊂ M_{X_{0},P̂}^{∗}(g_{0}) is obtained by pulling back the subset U ⊂ M_{X,P}^{∗} (g) of Theorem 5.8. □

Again, 𝒥̂ extends to a C^{∞} map on the larger parameter spaces 𝒯 and 𝒯^{0} and additional properties of 𝒥̂ are discussed in the next section.

5.2. Structure of the compactified moduli spaces. The bubbling ends of \overline{M}_{X_{0},k}^{u}(g_{0}) away from the diagonals are described in [D-K, §8.2]. We extend this description to neighbourhoods of points in the diagonals of the Uhlenbeck compactification. For related constructions and some further details, we refer to the papers of Taubes and Donaldson.

The proposition below is the basic result we require in order to parametrise neighbourhoods covering the ends of M_{X,k}^{∗}(g) away from the reducible connections. See also [D83, §III], [D86, §IV], and [T84b, p. 529] for various special cases of the statements below. The proof below is similar to arguments in the proofs of Proposition 5.17 & 5.22 [D-S] and Theorem 4.53 [D86, p. 316 & p. 325].

Proposition 5.10. Given the hypotheses of Theorem 5.8, the following hold:

(a) The approximate gluing map 𝒥′ : T/Γ → B_{X,k}^{∗} is a C^{∞} embedding,

(b) The gluing map 𝒥 : T/Γ → U ⊂ M_{X,k}^{∗}(g) is a diffeomorphism onto an open subset,

(c) The extended gluing map 𝒥 : 𝒯/Γ → 𝒰 ⊂ M_{X,k}^{∗}(g) is a C^{∞} submersion onto an open subset,

(d) The extended gluing map 𝒥 : 𝒯^{0}/Γ → 𝒰^{0} ⊂ M_{X,k}^{∗}(g) is a diffeomorphism onto an open subset.

Proof. (a) The proof is essentially the same as the argument required for (b) and so is omitted. (b) From Theorem 5.8, 𝒥 is a C^{∞} homeomorphism and so it is enough to show that 𝒥 is also an immersion, since T/Γ has dimension equal to that of M_{X,k}^{∗}(g). From the proof of Theorem 5.8, there is a C^{∞} Γ-equivariant gluing map 𝒥̃ : T → A_{X,k}^{∗}, t ↦ A(t). So, we first show that 𝒥̃ is an immersion and then conclude that the induced map on quotients is a diffeomorphism. The constant λ_{0} may be chosen as small as desired and in (a) and (b),the λ_{I} and x_{I} may be held fixed. Step 1. Definition of restriction maps. Choose cut-off functions ψ_{I}, as in §3.3, which are zero on the balls B_{Is}(b_{I}/2), B_{I_{+}}(b_{I_{+}}/2) and equal to 1 on the complement in X_{I} of the slightly larger balls B_{Is}(b_{I}), B_{I_{+}}(b_{I_{+}}). Define a map π_{X_{I}} : L^{2}Ω^{1}(X, ad P) → L^{2}Ω^{1}(X_{I}, ad P_{I}) by left multiplication with ψ_{I}, so that

(5.9)  ∥ω − π_{X_{I}} ω∥_{L^{2}(X_{I},g_{I})} = O(\overline{λ}),  ω ∈ Ω^{1}(X_{I}, g_{I}),

since ψ_{I} is equal to 1 on the complement of a set in X_{I} of g_{I}-volume O(\overline{λ}^{2}). Next, for I > 0, choose a cut-off function which is zero outside the annulus Ω_{Is} = Ω(x_{Is}, N^{−1}λ_{I}^{1/2}, Nλ_{I}^{1/2}) in X_{I} and which is equal to 1 on the slightly smaller annulus Ω(x_{Is}, \frac{1}{2}λ_{I}^{1/2}, 2λ_{I}^{1/2}) containing the supports of the derivatives of the cut-off functions γ_{I_{−}}, γ_{I}. Define a map π_{Ω_{I}} : L^{2}Ω^{1}(X, ad P) → L^{2}Ω^{1}(Ω_{Is}, ad P_{I}) by left multiplication with this cut-off function. Lastly, let Π = π_{0} ⊕_{I>0} (π_{X_{I}} ⊕ π_{Ω_{I}}) be the induced map

L^{2}Ω^{1}(X, ad P) ⟶ L^{2}Ω^{1}(X_{0}, ad P_{0}) ⨁_{I>0} (L^{2}Ω^{1}(X_{I}, ad P_{I}) ⊕ L^{2}Ω^{1}(Ω_{Is}, ad P_{I})) .

Step 2. Partial derivatives with respect to lower moduli parameters. We have C^{∞} Γ_{A_{I}} - equivariant maps ϑ̃_{I} : T_{A_{I}} → A_{X_{I},P_{I}}^{∗}, t_{I} ↦ A_{I}(t_{I}) given by the Kuranishi model. Let v be a tangent vector to T_{A_{I}}, i.e., suppose [v] ∈ H_{A_{I}}^{1} . Then Eq. (5.9) and the estimates of §5.4 give the following bounds for the differentials with respect to the lower moduli parameters:

(5.10)  ∥π_{X_{I}} D𝒥(v) − Dϑ̃_{I}(v)∥_{L^{2}(X_{I},g_{I})} = O(\overline{λ}^{1/2}).

The map ϑ̃_{I} is an immersion and so the range of Dϑ̃_{I} has dimension equal to dim H_{A_{I}}^{1} . For small enough \overline{λ}, Eq. (5.10) implies that the range of π_{X_{I}} D𝒥 also has dimension equal to dim H_{A_{I}}^{1} . Step 3. Partial derivatives with respect to gluing parameters. Let v be a tangent vector to Gl_{I}. The estimates of §5.5 give the following bounds for the differentials with respect to the gluing parameters:

(5.11)  ∥π_{Ω_{I}} D𝒥(v) − D𝒥′(v)∥_{L^{4}(X_{I},g_{I})} = O(\overline{λ}^{2}),

recalling that D𝒥′(v) is supported on Ω_{I}(\frac{1}{2}λ_{I}^{1/2}, 2λ_{I}^{1/2}). But from Proposition 3.28 we have

(5.12)  ∥D𝒥′(v)∥_{L^{4}(X_{I},g_{I})} ≥ c|v|,

for some constant c > 0 independent of \overline{λ}. In particular, the range of π_{Ω_{I}} D𝒥′ has dimension equal to dim Gl_{I}. So, for sufficiently small \overline{λ}, Eqs. (5.11) and (5.12) imply that the range of π_{Ω_{I}} D𝒥 also has dimension equal to dim Gl_{I}. Step 4. The quotient map. Combining these observations, we find that the range of ΠD𝒥 has dimension equal to dim H_{A_{0}} +∑_{I>0}(dim H_{A_{I}}^{1} +dim Gl_{I}) = dim T, so that ker ΠD𝒥 = 0 and 𝒥 is an immersion. From Theorem 5.8, the open subset Ũ ≡ 𝒥̃(T) in A_{X,k}^{∗} projects to an open subset U ≡ 𝒥(T) in M_{X,k}^{∗}(g) and composing 𝒥̃ with the projection A_{X,k}^{∗} → A_{X,k}^{∗}/G, we obtain a submersion 𝒥 : T → M_{X,k}^{∗}(g). The group Γ acts freely on T, 𝒥̃ is Γ-equivariant, and dim T/Γ = dim M_{X,k}^{∗}(g), and the gluing map descends to a diffeomorphism 𝒥 : T/Γ → M_{X,k}^{∗}(g), as required. (c) This follows from (b). For the derivatives with respect to λ_{I} or x_{I}, the cut-off functions required to define Π should be replaced by cut-offs with similar supports and which are fixed with respect to small variations in the scales and centres.

(d) This is similar to the proof of (c) and uses Proposition 3.5. □

In order to parametrise neighbourhoods of boundary points in \overline{M}_{X_{0},k}^{u}(g_{0}), we use the following corollary to Proposition 5.10.

Corollary 5.11. Given the hypotheses of Theorem 5.8, the following hold:

(a) The approximate gluing map 𝒥̂′ : T/Γ → B_{X_{0},k}^{∗} is a C^{∞} embedding,

(b) The gluing map 𝒥̂ : T/Γ → V ⊂ M_{X_{0},k}^{∗}(g_{0}) is a diffeomorphism onto an open subset,

(c) The extended gluing map 𝒥̂ : 𝒯/Γ → 𝒱 ⊂ M_{X_{0},k}^{∗}(g_{0}) is a C^{∞} submersion onto an open subset,

(d) The extended gluing map 𝒥̂ : 𝒯^{0}/Γ → 𝒱^{0} ⊂ M_{X_{0},k}^{∗}(g_{0}) is a diffeomorphism onto an open subset.

Taken together, Theorems 7.3.2 and 7.2.62 in [D-K] imply that if A is any g-anti-self-dual connection on a fixed G bundle P over the connected sum X and the necks Ω are all sufficiently pinched (so that \overline{λ} is small), then [A] lies in the image of the gluing map. The corresponding statement in our application is given below.

Theorem 5.12. Given the hypotheses of Theorem 5.8, then the following holds. Let {A_{α}}_{α=1}^{∞} be a sequence of connections on a G bundle P over the connected sum X = #_{I∈ℐ}X_{I} which are anti-self-dual with respect to the sequence of metrics {g_{α}}_{α=1}^{∞} determined by the sequences of scales \{λ_{Iα}\} with \overline{λ}_{α} → 0, a fixed neck width parameter N, sequences of points {x_{Iα}} converging to {x_{I}}, and frames in F X_{0}|_{x_{Iα}} converging to frames in F X_{0}|_{x_{I}}. Suppose the sequence {A_{α}}_{α=1}^{∞} is strongly convergent to (A_{I})_{I∈ℐ}, where A_{I} is a g_{I}-anti-self-dual connection over each summand X_{I}. For α_{0} sufficiently large, there exists a gluing neighbourhood 𝒰 such that [A_{α}] ∈ 𝒰, for all α ≥ α_{0}.

Proof. See [D-K, §7.3.1]. Theorem 4.18 implies that the sequence {A_{α}} is D_{q} convergent (for any 4 ≤ q < ∞) to (A_{I})_{I∈ℐ}. So, Theorem 5.8 implies that the points [A_{α}] are contained in a gluing neighbourhood 𝒰, for all α ≥ α_{0} if α_{0} is sufficiently large. □

Recall that Gl_{x_{I}} = SU(2) ≃ 𝕊^{3}, a copy of the standard three-sphere, and let \overline{Gl}_{x_{I}} be the closure of Gl_{x_{I}} × (0, λ_{0}) in the cone (Gl_{x_{I}} × [0, λ_{0}))/ ∼, where (ρ, 0) ∼ (ρ′, 0) if ρ, ρ′ ∈ Gl_{x_{I}}. Then, by analogy with [D-K, §8.2] and [D86, §V], we set

(5.13)  \overline{𝒯} ≡ T_{A_{0}} × ∏_{I∈ℐ}(T_{A_{I}} × B(x_{I}, r_{0}) × \overline{Gl}_{x_{I}})) ,

and likewise, define \overline{𝒯}^{0}. It is also convenient to define

(5.14)  ∂𝒯 ≡ \{t_{∞} = (t_{I}, y_{I}, ρ_{I}, λ_{I})_{I∈ℐ} ∈ \overline{𝒯} : λ_{I} = 0 for some I\},

where the 4-tuple (t_{I}, y_{I}, ρ_{I}, λ_{I}) above is replaced by t^{0}, if I = 0. The space ∂𝒯^{0} is defined similarly. Moreover, the gluing map 𝒥 has a natural definition on the boundary ∂𝒯. Suppose t_{∞} ∈ ∂𝒯 and let (λ_{1}, …, λ_{c}) denote the corresponding scales in Eq. (5.14) which have been set equal to zero. By cutting the edges with λ_{i} = 0, we may view the tree ℐ as a union of subtrees ∪_{i=1}^{c}ℐ^{i}. If t_{∞} ∈ ∂𝒯, we write t_{∞} = (t^{1}, …, t^{c}), with t^{i} ∈ 𝒯^{i}, and set

(5.15)  𝒥(t_{∞}) ≡ (𝒥(t^{1}), …, 𝒥(t^{c})),  t_{∞} ∈ 𝒯,

where each 𝒥(t^{i}) is an anti-self-dual connection over a connected sum Y_{i} = #_{I∈ℐ^{i}}X_{I}, say, and X = #_{I∈ℐ}X_{I} = #_{i=1}^{c}Y_{i}. The relationship between the gluing maps 𝒥 and 𝒥^{i} is explained by the continuity result below, which we just state in the special case X = X_{0}#X_{1}#X_{2}, for the sake of clarity. The argument required for this case carries over with no significant change to the more general cases just described.

Proposition 5.13. Let X = X_{0}#X_{1}#X_{2}, let Y = X_{0}#X_{1}, and let Y′′ = Y \ B(x_{1}, \frac{1}{2}λ_{1}^{1/2}). Assume the hypotheses of Theorem 5.8 and let 𝒥_{X}, 𝒥_{Y} be the gluing maps over the connected sums X and Y, respectively. Then there is an ε = ε(q) > 0 and a constant C = C(g_{0}, q, 𝒯) such that ∥𝒥_{X}(t)|_{Y′′} − 𝒥_{Y} ∥_{L^{q}(Y,g)} ≤ Cλ_{1}^{ε_{0}}.

The proof is similar to that of Proposition 7.2.64 [D-K] and the arguments in §5.3, and so is omitted. It now follows that 𝒥 extends continuously to \overline{𝒯}.

Proposition 5.14. Assume the hypotheses of Theorem 5.8. Let {t_{α}}_{α=1}^{∞} be a sequence in 𝒯 which converges to t_{∞} ∈ ∂𝒯. Then the sequence {𝒥(t_{α})}_{α=1}^{∞} converges strongly to 𝒥(t_{∞}).

Proof. Let {λ_{i}}_{i=1}^{c} denote the scales, determined by t_{∞}, which have been set equal to zero in Eq. (5.14). The points t_{α} ∈ 𝒯 are then naturally written as t_{α} = (t_{α}^{1}, …, t_{α}^{c}), with the sequences t_{α}^{i} converging to t^{i} ∈ 𝒯^{i}, say. According to Proposition 5.13, the sequence 𝒥(t_{α}) is then D_{q} convergent to (𝒥(t^{1}), …, 𝒥(t^{c})) and hence, strongly convergent by Theorem 4.18. □

It remains to show that M_{X_{0},k}(g_{0}) has a finite cover consisting of gluing neighbourhoods. Of course, away from the bubbling ends, the moduli space is covered by the standard Kuranishi charts. In addition, the geometry of these charts around the reducible connections has already been analysed in [G-P89], so our focus here is on the bubbling ends. Given any Uhlenbeck boundary point (A_{0}, x_{1}, …, x_{l}) ∈ \overline{M}_{X_{0},k}^{u}(g_{0}), where c_{2}(A_{0}) = k − l and each x_{i} has multiplicity 1, Theorem 8.2.3 [D-K] provides an open neighbourhood \overline{𝒱} of (A_{0}, x_{1}, …, x_{l}) in \overline{M}_{X_{0},k}^{u}(g_{0}), a parameter space 𝒯^{0}/Γ, and a gluing map 𝒥̂ giving a homeomorphism of 𝒯^{0}/Γ with 𝒱 = \overline{𝒱} ∩ M_{X_{0},k}^{∗}(g_{0}). Theorem 8.2.4 in [D-K] states that this

gluing map extends to a homeomorphism 𝒥̂ : \overline{𝒯}^{0}/Γ → \overline{𝒱}. Thus, away from the diagonals, the ends of \overline{M}_{X_{0},k}^{u}(g_{0}) are covered by gluing neighbourhoods. The generalisations below provide a covering of the ends of \overline{M}_{X_{0},k}^{u}(g_{0}), including the diagonal boundary points.

Theorem 5.15. Let (A_{0}, x_{1}, …, x_{m_{0}}) be a boundary point in \overline{M}_{X_{0},k}^{u}(g_{0}). Under the hypotheses of Theorem 5.8, there exist neighbourhoods \overline{𝒱} ⊂ \overline{M}_{X_{0},k}^{u}(g_{0}) of (A_{0}, x_{1}, …, x_{m_{0}}) and a parameter space 𝒯^{0} such that the following holds. If 𝒱 = \overline{𝒱} ∩ M_{X_{0},k}^{∗}(g_{0}), then the gluing map 𝒥̂ : 𝒯^{0}/Γ → 𝒱 is a diffeomorphism.

Proof. Suppose {[A_{α}]}_{α=1}^{∞} is a sequence in M_{X_{0},k}(g_{0}), converging weakly to the Uhlenbeck limit (A_{0}, x_{1}, …, x_{m_{0}}). Let {[Ǎ_{α}]}_{α=1}^{∞} be the corresponding strongly convergent sequence in M_{X,k}(g_{α}) with the bubble tree limit (A_{I}, x_{I})_{I∈ℐ}. Then Theorem 5.12 produces a gluing neighbourhood 𝒥(𝒯^{0}/Γ) = 𝒰 ⊂ M_{X,k}(g_{α}) and an α_{0} such that [Ǎ_{α}] ∈ 𝒰 for all α ≥ α_{0}. Let 𝒱 be the corresponding neighbourhood in M_{X_{0},k}(g_{0}). Then the conclusions follow from Corollary 5.11. □

Theorem 5.16. Given the hypotheses of Theorem 5.15, the gluing map 𝒥̂ extends to a homeomorphism of \overline{𝒯}^{0}/Γ with a neighbourhood \overline{𝒱} of (A_{0}, x_{1}, …, x_{m_{0}}) in \overline{M}_{X_{0},k}^{u}(g_{0}).

Proof. This follows from Proposition 5.14 and Theorem 5.16. □

Remark 5.17. So, every boundary point in \overline{M}_{X_{0},k}^{u}(g_{0}) has a neighbourhood constructible by gluing. Plainly, the same statement holds for boundary points in \overline{M}_{X_{0},k}^{τ}(g_{0}).

5.3. Derivatives with respect to scales and centres. The main purpose of this section is to obtain L^{2} estimates for the partial derivatives of the family of g_{0}-anti-self-dual connections Â with respect to the scales λ_{I} and centres x_{I}.

Unless noted otherwise, throughout this section and for the remainder of this article, we assume that p and q are Sobolev exponents satisfying the strict inequalities 2 < p < 4 and 4 < q < ∞, where q is determined by 1/p = 1/4+1/q. The constant λ_{0} > 0 is assumed small and may be decreased as needed. We use C = C(g_{0}, p, 𝒯) to denote constants which are independent of the points t = (t_{I}, ρ_{I}, x_{I}, λ_{I}) ∈ 𝒯. As usual, we abbreviate the derivative with respect to the centre parameters, p_{I}^{μ}∂/∂q_{I}^{μ} (where |p_{I}| ≤ 1) by ∂/∂p_{I}.

Denoting η ≡ −F^{+g}(A′) in Eq. (5.6), we have the following preliminary estimate for the derivatives of â with respect to the λ_{I} and x_{I} parameters.

Lemma 5.18. Let ξ and a = P ξ be as in Theorem 5.8, and assume that the conditions of that theorem hold. Then, for small enough λ_{0} > 0, there is a constant C = C(g_{0}, 𝒯) such that for any t ∈ 𝒯,

(a) ∥∂â/∂λ_{I}∥_{L^{2}(X_{0},g_{0})} ≤ C (∥\frac{∂P}{∂λ_{I}} ξ∥_{L^{2}(X,g)} + ∥∂ξ/∂λ_{I}∥_{L^{2}(X,g)} + \overline{λ}λ_{I}^{−1/2}),

(b) ∥∂â/∂p_{I}∥_{L^{2}(X_{0},g_{0})} ≤ C (∥\frac{∂P}{∂p_{I}} ξ∥_{L^{2}(X,g)} + ∥∂ξ/∂p_{I}∥_{L^{2}(X,g)} + \overline{λ}).

Proof. From Proposition 3.24, we have

∥\frac{∂â}{∂λ_{I}} ∥_{L^{2}(X_{0},g_{0})} ≤ C ∥\frac{∂a}{∂λ_{I}} ∥_{L^{2}(X,g)} + Cλ_{I}^{−1/2}∥a∥_{ℒ_{1}^{2}(X)} ,

where a = P ξ and \frac{∂P ξ}{∂λ_{I}} = \frac{∂P}{∂λ_{I}} ξ + P \frac{∂ξ}{∂λ_{I}}. The estimates of Proposition 5.5 and Theorem 5.6 then gives (a). The proof of (b) is similar. □

We now differentiate the g-anti-self-dual equation and obtain a priori estimates for the partial derivatives of ξ with respect to λ_{I} and x_{I}.

Lemma 5.19. Let ξ be as in Theorem 5.8 and assume that the conditions of that theorem hold. Then, for small enough λ_{0} > 0, there is a constant C = C(g_{0}, 𝒯) such that for any t ∈ 𝒯,

(a) ∥∂ξ/∂λ_{I}∥_{L^{2}(X,g)} ≤ C (1 + \overline{λ}^{2}λ_{I}^{−1/2} + \overline{λ}∥\frac{∂P}{∂λ_{I}} ξ∥_{L^{4}(X,g)}),

(b) ∥∂ξ/∂p_{I}∥_{L^{2}(X,g)} ≤ C (1 + \overline{λ}∥\frac{∂P}{∂p_{I}} ξ∥_{L^{4}(X,g)}).

Proof. Differentiating Eq. (5.6) with respect to λ_{I} gives

\frac{∂ξ}{∂λ_{I}} = \frac{∂η}{∂λ_{I}} − \frac{∂∗_{g}}{∂λ_{I}} (P ξ ∧ P ξ) − (\frac{∂P ξ}{∂λ_{I}} ∧ P ξ)^{+,g} − (P ξ ∧ \frac{∂P ξ}{∂λ_{I}} )^{+,g} .

The estimates of Lemma 3.14 and Proposition 5.5 imply that

∥\frac{∂ξ}{∂λ_{I}} ∥_{L^{2}} ≤ ∥\frac{∂η}{∂λ_{I}} ∥_{L^{2}} + C∥ξ∥_{L^{2}}^{2}λ_{I}^{−1/2} + C∥ξ∥_{L^{2}} (∥\frac{∂ξ}{∂λ_{I}} ∥_{L^{2}} + ∥\frac{∂P}{∂λ_{I}} ξ∥_{L^{4}}) .

Proposition 3.26 gives ∥∂η/∂λ_{I}∥_{L^{2}} ≤ C, while Theorem 5.6 gives ∥ξ∥_{L^{2}} ≤ C\overline{λ}. Thus, for λ_{0} small enough, we may assume C∥ξ∥_{L^{2}} ≤ 1/2. Part (a) then follows by combining the above estimates and rearranging, and the proof of (b) is similar. □

To complete our task, we need an estimate for the derivatives of P with respect to λ_{I} and x_{I}. Before proceeding, we first record some bounds for the derivatives of the cut-off functions β_{I} and γ_{I}. Suppose 1 ≤ p < ∞. From the definition of β_{I} there is a constant C = C(g_{I}, N, p) such that

(5.16)  |dβ_{I}|_{g_{I}} ≤ Cλ_{I}^{−1/2} on Ω_{I}, Ω_{Is} and ∥dβ_{I}∥_{L^{p}(X_{I},g_{I})} ≤ Cλ^{2/p−1/2}.

Second, for the derivatives of β_{J} with respect to λ_{I}, one has

(5.17) |∂β_{J}/∂λ_{I}|_{L^{∞}(X_{J} ,g_{J})} ≤ Cλ_{I}^{−1} and |∂dβ_{J}/∂λ_{I}|_{L^{∞}(X_{J} ,g_{J})} ≤ Cλ_{I}^{−3/2},
∥∂β_{J}/∂λ_{I}∥_{L^{p}(X_{J} ,g_{J})} ≤ λ_{I}^{2/p−1} and ∥∂dβ_{J}/∂λ_{I}∥_{L^{p}(X_{J} ,g_{J})} ≤ Cλ_{I}^{2/p−3/2},

for J = I_{−} or I, these derivatives being zero otherwise. Third, for the derivatives of β_{J} with respect to x_{I}, one has

(5.18) ∥∂β_{J}/∂p_{I}∥_{L^{∞}(X_{J} ,g_{J})} ≤ Cλ_{I}^{−1/2} and ∥∂dβ_{J}/∂p_{I}∥_{L^{∞}(X_{J} ,g_{J})} ≤ Cλ_{I}^{−1},
∥∂β_{J}/∂p_{I}∥_{L^{p}(X_{J} ,g_{J})} ≤ λ_{I}^{2/p−1/2} and ∥∂dβ_{J}/∂p_{I}∥_{L^{p}(X_{J} ,g_{J})} ≤ Cλ_{I}^{2/p−1},

for J = I_{−} or I, these derivatives being zero otherwise. The cut-off functions γ_{J} also satisfy the bounds of Eqs. (5.16), (5.17) and (5.18).

Proposition 5.20. For any 0 < δ < \frac{1}{2} and 2 < p < 4 defined by p = 4/(1 + 2δ), and small enough λ_{0}, there is a constant C = C(δ, g_{0}, 𝒯) such that for any t ∈ 𝒯 and ξ ∈ L^{p}Ω^{+,g}(X, ad P),

(a) ∥\frac{∂P}{∂λ_{I}} ξ∥_{L^{4}(X,g)} ≤ Cλ_{I}^{−1/2−δ}∥ξ∥_{L^{p}(X,g)},

(b) ∥\frac{∂P}{∂p_{I}} ξ∥_{L^{4}(X,g)} ≤ Cλ_{I}^{−δ}∥ξ∥_{L^{p}(X,g)}.

Proof. (a) As P = Q(1 + R)^{−1}, we first obtain operator bounds for ∂Q/∂λ_{I}, ∂R/∂λ_{I}, and then deduce an operator bound for ∂P/∂λ_{I}. Step 1. Estimate for ∂Q/∂λ_{I}. Recall that Qξ = ∑_{J} Q_{J}ξ, where Q_{J} = β_{J}P_{J}γ_{J} is independent of λ_{I} for J ≠ I_{−}, I, and so

\frac{∂Q}{∂λ_{I}} = \frac{∂Q_{I_{−}}}{∂λ_{I}} + \frac{∂Q_{I}}{∂λ_{I}} , where \frac{∂Q_{I}}{∂λ_{I}} = \frac{∂β_{I}}{∂λ_{I}} P_{I}γ_{I} + β_{I}P_{I} \frac{∂c_{I}}{∂λ_{I}} ,

with the analogous expression for ∂Q_{I_{−}}/∂λ_{I}. Choose 4 < q, q_{1} < ∞ and 2 < p, p_{1} < 4 by setting

(5.19)  p = 4/(1 + 2δ) and q = 4/(1 − 2δ)
1/p = 1/4 + 1/q_{1} and 1/2 = 1/p_{1} + 1/q_{1},

and observe that 1/4 = 1/q + 1/q_{1} and 1/2 = 1/p + 1/q, while 2/p = 1/2 + δ and 2/q = 1/2 − δ. Applying Hölder’s inequality, the operator bounds for P_{I} of Lemma 5.1, and the fact that ∥∂β_{I}/∂λ_{I}∥_{L^{q}} and ∥∂γ_{I}/∂λ_{I}∥_{L^{q}} are bounded by Cλ_{I}^{2/q−1} from Eq. (5.17), we find

∥\frac{∂Q_{I}}{∂λ_{I}} ξ∥_{L^{4}} ≤ C ∥\frac{∂β_{I}}{∂λ_{I}} ∥_{L^{q}} ∥ξ∥_{L^{p}} + C ∥\frac{∂γ_{I}}{∂λ_{I}} ∥_{L^{q}} ∥ξ∥_{L^{p}} ≤ Cλ_{I}^{2/q−1}∥ξ∥_{L^{p}}.

Combining the above estimate with the analogous bound for the ∂Q_{I_{−}}/∂λ_{I} term, we see that

(5.20)  ∥\frac{∂Q}{∂λ_{I}} ξ∥_{L^{4}} ≤ Cλ_{I}^{2/q−1}∥ξ∥_{L^{p}},

completing Step 1. Step 2. Estimate for ∂R/∂λ_{I}. We have R = d_{A′}^{+,g}Q − 1 on X and so differentiating with respect to λ_{I} gives

\frac{∂R}{∂λ_{I}} ξ = \frac{∂∗_{g}}{∂λ_{I}} d_{A′}Qξ + [\frac{∂A′}{∂λ_{I}} , Qξ]^{+,g} + d_{A′}^{+,g} \frac{∂Q}{∂λ_{I}} ξ.

Using our L^{∞} bound for ∂∗_{g} /∂λ_{I} of Lemma 3.14, the L^{4} bound for ∂A′/∂λ_{I} of Proposition 3.25, and the operator norm bounds for Q of Lemma 5.3, we see that

(5.21)  ∥\frac{∂R}{∂λ_{I}} ξ∥_{L^{2}} ≤ Cλ_{I}^{−1/2}∥d_{A′}Qξ∥_{L^{2}} + C∥ξ∥_{L^{2}} + ∥d_{A′}^{+,g} \frac{∂Q}{∂λ_{I}} ξ∥_{L^{2}} .

For the d_{A′}Q term above, note that d_{A′}Qξ = ∑_{J} d_{A′_{J}} Q_{I}ξ and writing A′_{J} = A_{J} + a_{J} over X′_{J}, we have

d_{A′_{J}} Q_{J}ξ = dβ_{J} ∧ P_{J}γ_{J}ξ + β_{J}d_{A_{J}} P_{J}γ_{J}ξ + β_{J}[a_{J}, P_{J}γ_{J}ξ].

Using the bounds ∥dβ_{J}∥_{L^{4}} ≤ C of Eq. (5.16), ∥a_{J}∥_{L^{4}} ≤ C\overline{λ} of Lemma 3.9, Hölder’s inequality, and the operator bounds for P_{J} of Lemma 5.1, we find that

(5.22)  ∥d_{A′}Qξ∥_{L^{2}} ≤ C∥ξ∥_{L^{2}}.

For the d_{A′}^{+,g}∂Q/∂λ_{I} term, note that

d_{A′}^{+,g} \frac{∂Q}{∂λ_{I}} = d_{A′_{I_{−}}}^{+,g} \frac{∂Q_{I_{−}}}{∂λ_{I}} + d_{A′_{I}}^{+,g} \frac{∂Q_{I}}{∂λ_{I}} ,

GEOMETRY OF MODULI SPACE ENDS 59

where, using d_{A_{I}}^{+,g_{I}} P_{I} = 1 and β_{I} = 1 on supp γ_{I}, we have

d_{A′_{I}}^{+,g} \frac{∂Q_{I}}{∂λ_{I}} ξ = (d\frac{∂β_{I}}{∂λ_{I}} ∧ P_{I}γ_{I}ξ)^{+,g} + \frac{∂β_{I}}{∂λ_{I}} [a_{I}, P_{I}γ_{I}ξ]^{+,g}

+ (dβ_{I} ∧ P_{I} \frac{∂γ_{I}}{∂λ_{I}} ξ)^{+,g} + \frac{∂γ_{I}}{∂λ_{I}} ξ + β_{I} [a_{I}, P_{I} \frac{∂γ_{I}}{∂λ_{I}} ξ]^{+,g}

+ \frac{1}{2} \frac{∂β_{I}}{∂λ_{I}} (∗_{g} − ∗_{g_{I}})d_{A_{I}} P_{I}γ_{I}ξ + \frac{1}{2}β_{I}(∗_{g} − ∗_{g_{I}})d_{A_{I}} P_{I} \frac{∂γ_{I}}{∂λ_{I}} ξ,

with the analogous expression for d_{A′_{I_{−}}}^{+,g} ∂Q_{I_{−}}/∂λ_{I}. From Lemma 3.14 and Lemma 5.1, we see that

∥d_{A′_{I}}^{+,g} \frac{∂Q_{I}}{∂λ_{I}} ξ∥_{L^{2}} ≤ C ∥d\frac{∂β_{I}}{∂λ_{I}} ∥_{L^{p_{1}}} ∥ξ∥_{L^{p}} + C ∥\frac{∂β_{I}}{∂λ_{I}} ∥_{L^{q}} ∥a_{I}∥_{L^{4}}∥ξ∥_{L^{p}}

+ C∥dβ_{I}∥_{L^{4}} ∥\frac{∂γ_{I}}{∂λ_{I}} ∥_{L^{q}} ∥ξ∥_{L^{p}} + ∥\frac{∂γ_{I}}{∂λ_{I}} ∥_{L^{q}} ∥ξ∥_{L^{p}} + C∥a_{I}∥_{L^{4}} ∥\frac{∂γ_{I}}{∂λ_{I}} ∥_{L^{q}} ∥ξ∥_{L^{p}}

+ C\overline{λ} ∥\frac{∂β_{I}}{∂λ_{I}} ∥_{L^{q}} ∥ξ∥_{L^{p}} + C\overline{λ} ∥\frac{∂γ_{I}}{∂λ_{I}} ∥_{L^{q}} ∥ξ∥_{L^{p}},

Now ∥a_{I}∥_{L^{4}} ≤ C\overline{λ} by Lemma 3.9, and from Eq. (5.17), we have that ∥∂β_{I}/∂λ_{I}∥_{L^{q}} and ∥∂γ_{I}/∂λ_{I}∥_{L^{q}} are bounded by Cλ_{I}^{2/q−1}. Hence,

∥d_{A′_{I}}^{+,g} \frac{∂Q_{I}}{∂λ_{I}} ξ∥_{L^{2}} ≤ Cλ_{I}^{2/q−1}∥ξ∥_{L^{p}},

with the analogous bound for the d_{A′_{I_{−}}}^{+,g} ∂Q_{I_{−}}/∂λ_{I} term. Therefore,

(5.23) ∥d_{A′}^{+,g} \frac{∂Q}{∂λ_{I}} ξ∥_{L^{2}} ≤ Cλ_{I}^{2/q−1}∥ξ∥_{L^{p}}.

Combining the above inequalities and noting that ∥ξ∥_{L^{2}(X,g)} ≤ C∥ξ∥_{L^{p}(X,g)}, we have

(5.24) ∥\frac{∂R}{∂λ_{I}} ξ∥_{L^{4}(X,g)} ≤ Cλ_{I}^{2/q−1}∥ξ∥_{L^{p}},

which completes Step 2. Step 3. Estimate for ∂P/∂λ_{I}. Differentiating P = Q(1 + R)^{−1} with respect to λ_{I} gives

\frac{∂P}{∂λ_{I}} = \frac{∂Q}{∂λ_{I}} (1 + R)^{−1} − Q(1 + R)^{−1} \frac{∂R}{∂λ_{I}} (1 + R)^{−1},

and thus applying the bounds from Steps 1 and 2, we have

∥\frac{∂P}{∂λ_{I}} ξ∥_{L^{4}(X,g)} ≤ Cλ_{I}^{2/q−1}∥ξ∥_{L^{p}(X,g)},

which yields (a) since 2/q −1 = −1/2−δ. For (b), the strategy of (a) shows that ∥\frac{∂Q}{∂p_{I}} ξ∥_{L^{4}}

and ∥\frac{∂R}{∂p_{I}} ξ∥_{L^{4}(X,g)} are bounded by Cλ_{I}^{2/q−1/2}∥ξ∥_{L^{p}}, giving

(5.25)  ∥\frac{∂P}{∂p_{I}} ξ∥_{L^{4}(X,g)} ≤ Cλ_{I}^{2/q−1/2}∥ξ∥_{L^{p}},

and so (b) follows. □

As is readily verified, Lemma 5.19 and Proposition 5.20 then provide the the following estimates for the derivatives of ξ and a with respect to λ_{I} and x_{I}:

Corollary 5.21. Let ξ and a = P ξ be as in Theorem 5.8 and assume that the conditions of that theorem hold. Then, for small enough λ_{0} > 0, there is a constant C = C(δ, g_{0}, 𝒯) such that for any t ∈ 𝒯,

(a) ∥∂ξ/∂λ_{I}∥_{L^{2}(X,g)} ≤ C(1 + \overline{λ}^{3/2+δ}λ_{I}^{−1/2−δ}),

(b) ∥∂ξ/∂p_{I}∥_{L^{2}(X,g)} ≤ C(1 + \overline{λ}^{3/2+δ}λ_{I}^{−δ}),

(c) ∥∂a/∂λ_{I}∥_{L^{2}(X,g)} ≤ C(1 + \overline{λ}^{1/2+δ}λ_{I}^{−1/2−δ}),

(d) ∥∂a/∂p_{I}∥_{L^{2}(X,g)} ≤ C(1 + \overline{λ}^{1/2+δ}λ_{I}^{−δ}).

With bounds for the derivatives of ξ and P with respect to λ_{I} and x_{I} at hand, we obtain our final estimates for the derivatives of the anti-self-dual connections A and Â. Since A = A′ + a and combining Proposition 3.25 and Corollary 5.21, we have:

Corollary 5.22. Assume that the conditions of Theorem 5.8 hold. Then, for any 0 < δ < 1/2 and small enough λ_{0} > 0, there is a constant C = C(δ, g_{0}, 𝒯) such that for any t ∈ 𝒯, the following bounds hold:

(a) ∥∂A/∂λ_{I}∥_{L^{2}(X,g)} ≤ C(1 + \overline{λ}^{1/2+δ}λ_{I}^{−1/2−δ}),

(b) ∥∂A/∂p_{I}∥_{L^{2}(X,g)} ≤ C(1 + \overline{λ}^{1/2+δ}λ_{I}^{−δ}).

Theorem 5.23. Assume that the conditions of Theorem 5.8 hold. Then, for any 0 < δ < 1/2 and small enough λ_{0} > 0, there is a constant C = C(δ, g_{0}, 𝒯) such that for any t ∈ 𝒯, the following bounds hold:

(a) ∥∂â/∂λ_{I}∥_{L^{2}(X_{0},g_{0})} ≤ C(1 + \overline{λ}^{1/2+δ}λ_{I}^{−1/2−δ}),

(b) ∥∂Â/∂λ_{I}∥_{L^{2}(X_{0},g_{0})} ≤ C(1 + \overline{λ}^{1/2+δ}λ_{I}^{−1/2−δ}),

(c) ∥∂â/∂p_{I}∥_{L^{2}(X_{0},g_{0})} ≤ C(1 + \overline{λ}^{1/2+δ}λ_{I}^{−δ}), (d) ∥∂Â/∂p_{I}∥_{L^{2}(X_{0},g_{0})} ≤ C(1 + \overline{λ}^{1/2+δ}λ_{I}^{−δ}).

Proof. Using the bound ∥ξ∥_{L^{p}} ≤ C\overline{λ}^{2/p} of Theorem 5.6, the equality 2/p = 1/2 + δ, the L^{2} estimate for ∂â/∂λ_{I} in Lemma 5.18, the L^{2} estimate for ∂ξ/∂λ_{I} in Corollary 5.21, and the operator estimate for ∂P/∂λ_{I} in Proposition 5.20, we obtain

∥\frac{∂â}{∂λ_{I}} ∥_{L^{2}(X_{0},g_{0})} ≤ C (\overline{λ}^{1/2+δ}λ_{I}^{−1/2−δ} + 1 + \overline{λ}^{3/2+δ}λ_{I}^{−1/2−δ} + \overline{λ}^{1/2+δ}λ_{I}^{−1/2}) ,

which gives (a). Then (b) follows from (a) and the estimate ∥∂Â′/∂λ_{I}∥_{L^{2}(X_{0},g_{0})} ≤ C of Proposition 3.27. The proofs of (c) and (d) are similar. □

5.4. Derivatives with respect to lower moduli. In this section we obtain estimates for the derivatives of the family of g_{0}-anti-self-dual connections Â with respect to the lower moduli parameters t_{I} ∈ T_{A_{I}}. Just as in §5.3, the strategy is to use the g-anti-self-dual equation of Eq. (5.6), together with its derivatives with respect to the t_{I} parameters, to first obtain estimates for the derivatives of a and ξ, and then the required derivatives of â and Â′. The Sobolev exponents p, q are fixed so that 2 ≤ p < 4 and 4 ≤ q < ∞, where q is determined by 1/p = 1/4 + 1/q. We have the following preliminary estimates for the derivatives of ξ and a.

Lemma 5.24. Let ξ and a = P ξ be as in Theorem 5.8, and assume that the conditions of that theorem hold. Then, for small enough λ_{0} > 0, there is a constant C = C(g_{0}, p, 𝒯) such that for any t ∈ 𝒯,

(a) ∥∂a/∂t_{I}∥_{L^{p}(X,g)} ≤ C∥∂ξ/∂t_{I}∥_{L^{p}(X,g)} + ∥\frac{∂P}{∂t_{I}} ξ∥_{L^{p}(X,g)},

(b) ∥∂ξ/∂t_{I}∥_{L^{p}(X,g)} ≤ C (\overline{λ}^{2/p−1/2} + \overline{λ}^{2/p}∥\frac{∂P}{∂t_{I}} ξ∥_{L^{4}(X,g)}).

Proof. Differentiating Eq. (5.6) with respect to t_{I} gives

\frac{∂ξ}{∂t_{I}} = \frac{∂η}{∂t_{I}} − (\frac{∂P ξ}{∂t_{I}} ∧ P ξ)^{+,g} − (P ξ ∧ \frac{∂P ξ}{∂t_{I}} )^{+,g} ,
\frac{∂P ξ}{∂t_{I}} = \frac{∂P}{∂t_{I}} ξ + P \frac{∂ξ}{∂t_{I}} .

The proofs of (a) and (b) are then similar to those of Lemmas 5.18 and 5.19. □

Thus, an operator estimate for ∂P/∂t_{I} is required. Since P = Q(1 + R)^{−1}, we have

(5.26)  \frac{∂P}{∂t_{I}} = \frac{∂Q}{∂t_{I}} (1 + R)^{−1} − Q(1 + R)^{−1} \frac{∂R}{∂t_{I}} (1 + R)^{−1}.

We recall that P_{I} = d_{A_{I}}^{∗,g_{I}} G_{A_{I}}^{+,g_{I}}. Differentiating with respect to t_{I}, we obtain

\frac{∂P_{I}}{∂t_{I}} = \frac{∂d_{A_{I}}^{∗,g_{I}}}{∂t_{I}} G_{A_{I}}^{+,g_{I}} − d_{A_{I}}^{∗,g_{I}} G_{A_{I}}^{+,g_{I}} \frac{∂Δ_{A_{I}}^{+,g_{I}}}{∂t_{I}} G_{A_{I}}^{+,g_{I}} .

The derivatives of d_{A_{I}}^{+,g_{I}} and d_{A_{I}}^{∗,g_{I}} with respect to t_{I} are given by

\frac{∂d_{A_{I}}^{+,g_{I}}}{∂t_{I}} ω = [\frac{∂A_{I}}{∂t_{I}} , ω]^{+,g_{I}} = [\frac{∂A_{I}}{∂t_{I}} , ·]^{+,g_{I}} ω,
\frac{∂d_{A_{I}}^{∗,g_{I}}}{∂t_{I}} ξ = − ∗ [\frac{∂A_{I}}{∂t_{I}} , ∗ξ] = [\frac{∂A_{I}}{∂t_{I}} , ·]^{∗} ξ,

for any ω ∈ Ω^{1}(X_{I}, ad P_{I}) and ξ ∈ Ω^{+,g_{I}} (X_{I}, ad P_{I}). Therefore,

\frac{∂Δ_{A_{I}}^{+,g_{I}}}{∂t_{I}} = [\frac{∂A_{I}}{∂t_{I}} , ·]^{+,g_{I}} d_{A_{I}}^{∗,g_{I}} + d_{A_{I}}^{+,g_{I}} [\frac{∂A_{I}}{∂t_{I}} , ·]^{∗} ,

and so we find that

(5.27)  \frac{∂P_{I}}{∂t_{I}} = (1 − P_{I}d_{A_{I}}^{+,g_{I}}) [\frac{∂A_{I}}{∂t_{I}} , ·]^{∗} G_{A_{I}}^{+,g_{I}} − P_{I} [\frac{∂A_{I}}{∂t_{I}} , ·]^{+,g_{I}} P_{I}.

Note that 1−P_{I}d_{A_{I}}^{+,g_{I}} is a bounded (L^{q}, L^{q}) operator on Ω^{+,g_{I}} (X_{I}, ad P_{I}) by the Calderon-Zygmund theory.

Lemma 5.25. There is a constant C = C(g_{0}, p, 𝒯) such that for any t ∈ 𝒯,

(a) ∥\frac{∂P_{I}}{∂t_{I}} ξ∥_{L^{q}(X_{I},g_{I})} ≤ C∥ξ∥_{L^{p}(X_{I},g_{I})}, for ξ ∈ L^{p}Ω^{+,g_{I}} (X_{I}, g_{I}),

(b) ∥\frac{∂Q}{∂t_{I}} ξ∥_{L^{q}(X,g)} ≤ C∥ξ∥_{L^{p}(X,g)}, for ξ ∈ L^{p}Ω^{+,g}(X, g).

Proof. Since 1 − P_{I}d_{A_{I}}^{+,g_{I}} is bounded on L^{q}(X_{I}, g_{I}), then Eq. (5.27) and the Hölder inequalities show that

∥\frac{∂P_{I}}{∂t_{I}} ξ∥_{L^{q}} ≤ ∥\frac{∂A_{I}}{∂t_{I}} ∥_{L^{∞}} ∥G_{A_{I}}^{+,g_{I}} ξ∥_{L^{q}} + C ∥\frac{∂A_{I}}{∂t_{I}} ∥_{L^{4}} ∥P_{I}ξ∥_{L^{q}}.

But G_{A_{I}}^{+,g_{I}} and P_{I} are bounded (L^{p}, L^{q}) operators and noting that the family A_{I}(t_{I}) is smoothly parametrised by t_{I} ∈ T_{A_{I}}, we obtain (a). Now Q_{I} = β_{I}P_{I}γ_{I} and Q = ∑_{I} Q_{I}, so inequality (b) follows. □

It remains to estimate the derivative of R with respect to t_{I}.

Lemma 5.26. There exists a constant C = C(g_{0}, p, 𝒯) such that for any t ∈ 𝒯 and ξ ∈ Ω^{+,g}(X, g), we have ∥\frac{∂R}{∂t_{I}} ξ∥_{L^{p}(X,g)} ≤ C∥ξ∥_{L^{p}(X,g)}.

Proof. We recall that R = d_{A′}^{+,g}Q − 1 over X and R = d_{A′_{I}}^{+,g}Q_{I} − 1 over X_{I}. Writing A′_{I} = A_{I} + a_{I}, we find that

R = dβ_{I} ∧ P_{I}γ_{I} + β_{I}d_{A_{I}}^{+,g_{I}} P_{I}γ_{I} + β_{I}[a_{I}, · ]^{+,g}P_{I}γ_{I} + \frac{1}{2}(∗_{g} − ∗_{g_{I}})β_{I}d_{A_{I}} P_{I} − 1.

Noting that d_{A_{I}}^{+,g_{I}} P_{I} = 1 and differentiating with respect to t_{I}, we have

(5.28)  \frac{∂R}{∂t_{I}} ξ = dβ_{I} ∧ \frac{∂P_{I}}{∂t_{I}} γ_{I}ξ + β_{I} [\frac{∂a_{I}}{∂t_{I}} , P_{I}γ_{I}ξ]^{+,g} + β_{I} [a_{I}, \frac{∂P_{I}}{∂t_{I}} γ_{I}ξ]^{+,g}

+ \frac{1}{2}(∗_{g} − ∗_{g_{I}})β_{I} [\frac{∂A_{I}}{∂t_{I}} , P_{I}γ_{I}ξ] + \frac{1}{2}(∗_{g} − ∗_{g_{I}})β_{I}d_{A_{I}} \frac{∂P_{I}}{∂t_{I}} γ_{I}ξ,

and so

(5.29)  ∥\frac{∂R}{∂t_{I}} ξ∥_{L^{p}} ≤ C∥dβ_{I}∥_{L^{4}} ∥\frac{∂P_{I}}{∂t_{I}} γ_{I}ξ∥_{L^{q}} + C ∥\frac{∂a_{I}}{∂t_{I}} ∥_{L^{4}} ∥P_{I}γ_{I}ξ∥_{L^{q}}

+ C∥a_{I}∥_{L^{4}} ∥\frac{∂P_{I}}{∂t_{I}} γ_{I}ξ∥_{L^{q}} + C\overline{λ} ∥\frac{∂A_{I}}{∂t_{I}} ∥_{L^{4}} ∥P_{I}γ_{I}ξ∥_{L^{q}}

+ C\overline{λ} ∥d_{A_{I}} \frac{∂P_{I}}{∂t_{I}} γ_{I}ξ∥_{L^{p}} ,

where a_{I} = (ψ_{I} − 1)σ_{I}^{∗}A_{I} and ∂a_{I}/∂t_{I} = (ψ_{I} − 1)σ_{I}^{∗}∂A_{I}/∂t_{I}. Aside from the self-dual projection and factor β_{I}, the last term on the right-hand side of Eq. (5.28) is given by

d_{A_{I}} \frac{∂P_{I}}{∂t_{I}} γ_{I}ξ = −d_{A_{I}} ∗ [\frac{∂A_{I}}{∂t_{I}} , G_{A_{I}}^{+,g_{I}} γ_{I}ξ] + d_{A_{I}} P_{I}d_{A_{I}}^{+,g_{I}} ∗ [\frac{∂A_{I}}{∂t_{I}} , G_{A_{I}}^{+,g_{I}} γ_{I}ξ]

− d_{A_{I}} P_{I} [\frac{∂A_{I}}{∂t_{I}} , P_{I}γ_{I}ξ]^{+,g} .

Since P_{I} is a bounded operator from L^{p} to L_{1}^{p} and using the bounded inclusion L_{1}^{p} → L^{q}, we see that

∥d_{A_{I}} \frac{∂P_{I}}{∂t_{I}} γ_{I}ξ∥_{L^{p}} ≤ C ∥d_{A_{I}} ∗ [\frac{∂A_{I}}{∂t_{I}} , G_{A_{I}}^{+,g_{I}} γ_{I}ξ]∥_{L^{p}} + ∥[\frac{∂A_{I}}{∂t_{I}} , P_{I}γ_{I}ξ]∥_{L^{p}}

≤ C ∥\frac{∂A_{I}}{∂t_{I}} ∥_{L^{∞}} ∥G_{A_{I}}^{+,g_{I}} γ_{I}ξ∥_{L^{p}} + ∥∇^{A_{I}} \frac{∂A_{I}}{∂t_{I}} ∥_{L^{∞}} ∥G_{A_{I}}^{+,g_{I}} γ_{I}ξ∥_{L^{p}}

+ C ∥\frac{∂A_{I}}{∂t_{I}} ∥_{L^{∞}} ∥∇^{A_{I}} G_{A_{I}}^{+,g_{I}} γ_{I}ξ∥_{L^{p}} + ∥\frac{∂A_{I}}{∂t_{I}} ∥_{L^{4}} ∥P_{I}γ_{I}ξ∥_{L^{q}}.

Since the family A_{I}(t) is smoothly parametrised by t_{I} ∈ T_{A_{I}} and as G_{A_{I}}^{+,g_{I}} is a bounded operator from L^{p} to L_{2}^{p}, we have

(5.30)  ∥d_{A_{I}} \frac{∂P_{I}}{∂t_{I}} γ_{I}ξ∥_{L^{p}} ≤ C∥ξ∥_{L^{p}}.

Eqs. (5.29), (5.30) and Lemma 5.25 then yield the required bound for ∂R/∂t_{I}. □

Thus, Eq. (5.26), together with Lemmas 5.25 and 5.26, provides an estimate for the derivative of P with respect to t_{I}:

Proposition 5.27. There is a constant C = C(g_{0}, p, 𝒯) such that for any t ∈ 𝒯 and ξ ∈ L^{p}Ω^{+,g}(X, ad P), we have ∥\frac{∂P}{∂t_{I}} ξ∥_{L^{q}(X,g)} ≤ C∥ξ∥_{L^{p}(X,g)}.

This leads to our final estimates for the derivatives of ξ and a with respect to t_{I}.

Corollary 5.28. Let ξ and a = P ξ be as in Theorem 5.8, and assume that the conditions of that theorem hold. Then, for small enough λ_{0} > 0, there is a constant C = C(g_{0}, p, 𝒯) such that for any t ∈ 𝒯,

(a) ∥∂ξ/∂t_{I}∥_{L^{p}(X,g)} ≤ C\overline{λ}^{2/p−1/2},

(b) ∥∂a/∂t_{I}∥_{L^{p}(X,g)} ≤ C\overline{λ}^{2/p−1/2}.

Proof. Inequality (a) follows from Lemma 5.24 and Proposition 5.27, since ∥ξ∥_{L^{p}} ≤ C\overline{λ}^{2/p} by Theorem 5.6. Inequality (b) then follows from (a) and Lemma 5.24. □

By combining Proposition 3.30 and Corollary 5.28 we obtain an estimate for the derivatives of the connections A = A′ + a over X:

Corollary 5.29. Assume that the conditions of Theorem 5.8 hold. Then, for any 2 ≤ p < 4 and sufficiently small λ_{0} > 0, there is a constant C = C(g_{0}, p, 𝒯) such that for any t ∈ 𝒯,

(a) ∥∂A/∂t_{I} − ∂A_{I}/∂t_{I}∥_{L^{p}(X′′_{I},g_{I})} ≤ C\overline{λ}^{2/p−1/2},

(b) ∥∂A/∂t_{I}∥_{L^{p}(X,g)} ≤ C.

We now come to the main result of this section.

Theorem 5.30. Assume that the conditions of Theorem 5.8 hold. Then, for any 2 ≤ p < 4 and sufficiently small λ_{0} > 0, there is a constant C = C(g_{0}, p, 𝒯) such that for any t ∈ 𝒯,

(a) ∥∂â/∂t_{I}∥_{L^{p}(X_{0},g_{0})} ≤ C\overline{λ}^{2/p−1/2}

(b) ∥∂Â/∂t_{I}∥_{L^{p}(X_{0},g_{0})} ≤ C.

Proof. Let U ≡ f_{0}^{−1} ⋯ f_{I}^{−1}(X′_{I}) ⊂ X_{0} and note that \frac{∂â}{∂t_{I}} = ∑_{I} f_{0}^{∗} ⋯ f_{I}^{∗} \frac{∂a}{∂t_{I}} on U. Now Lemma 3.19 gives

∥\frac{∂â}{∂t_{I}} ∥_{L^{p}(X_{0},g_{0})} ≤ C ∑_{I} ∥f_{0}^{∗} ⋯ f_{I}^{∗} \frac{∂a}{∂t_{I}} ∥_{L^{p}(X_{0},g_{0})} ≤ C ∥\frac{∂a}{∂t_{I}} ∥_{L^{p}(X,g)} ,

and so Part (a) follows from Corollary 5.28. Then Part (b) follows from (a), the estimate ∥∂Â′/∂t_{I}∥_{L^{p}(X_{0},g_{0})} ≤ C of Proposition 3.32. □

5.5. Derivatives with respect to bundle gluing parameters. We obtain estimates for the partial derivatives of the family of g_{0}-anti-self-dual connections Â(t) with respect to the bundle gluing parameters ρ_{I} ∈ Gl_{I}. The Sobolev exponents p, q are fixed so that 2 ≤ p < 4, with 4 ≤ q < ∞ determined by 1/4 + 1/q = 1/p. We first recall the estimate of Donaldson and Kronheimer for the derivative of a = P ξ with respect to the gluing parameters ρ_{I}. As described in §3.8, we work with an equivalent family of g-anti-self-dual connections A = A′ +a on a fixed bundle P. Thus, considering only the gluing parameters, we have a diffeomorphism B_{𝔤} ∋ v → A(ρ̄_{I}, v) ∈ A_{X,P}^{∗} (where B_{𝔤} is the unit ball in 𝔤), giving a family of C^{∞} connections on a fixed bundle P = P (ρ̄_{I}), as in Eq. (3.58). Here, B_{𝔤} ∋ v → ρ_{I}(v) = ρ̄_{I} exp(v) ∈ Gl_{I} is a coordinate chart centred at ρ̄_{I} ∈ Gl_{I}, as in Eq. (3.57). This understood, one has the following bounds.

Proposition 5.31. [D-K, p. 303] Let a be as in Theorem 5.8, and assume that the conditions of that theorem hold. Then, for small enough λ_{0} > 0, there is a constant C = C(g_{0}, p, 𝒯) such that for any t ∈ 𝒯, ∥∂a/∂v∥_{L^{q}(X,g)} ≤ C\overline{λ}^{2/p+1}.

Proof. The proof in [D-K] deals only with single connected sums X = X_{0}#X_{1}, but the argument adapts without significant change to the general case of multiple connected sums #_{I∈ℐ}X_{I}. Likewise, the assumptions in [D-K] that Γ_{I} = 1 and H_{A_{I}}^{0} = H_{A_{I}}^{1} = 0, for all I, do not affect the relevant estimates. □

Corollary 5.32. Let A be as in Theorem 5.8, and assume that the conditions of that theorem hold. Then, for small enough λ_{0} > 0, there is a constant C = C(g_{0}, p, 𝒯) such that for any t ∈ 𝒯, ∥∂A/∂v∥_{L^{p}(X,g)} ≤ C\overline{λ}^{2/p−1/2}.

Proof. Combine Propositions 3.28 and 5.31. □

Moreover, we have the following estimates for the derivatives of the g_{0}-anti-self-dual connections Â = Â′ + â on the fixed bundle P̂ over X_{0}.

Theorem 5.33. Assume that the conditions of Theorem 5.8 hold. Then, for small enough λ_{0} > 0, there is a constant C = C(g_{0}, p, 𝒯) such that for any t ∈ 𝒯,

(a) ∥∂â/∂v∥_{L^{p}(X,g)} ≤ C\overline{λ}^{2/p+1},

(b) ∥∂Â/∂v∥_{L^{p}(X,g)} ≤ C\overline{λ}^{2/p−1/2}.

Proof. Since â = f_{0}^{∗} ⋯ f_{I}^{∗}a on U ≡ f_{0}^{−1} ∘ ⋯ ∘ f_{I}^{−1}(X′_{I}), Lemma 3.19 gives

∥f_{0}^{∗} ⋯ f_{I}^{∗} \frac{∂a}{∂v}∥_{L^{p}(U,g_{0})} ≤ C ∥\frac{∂a}{∂v}∥_{L^{p}(X′_{I},g_{I})} ,

and Proposition 5.31 gives (a). Similarly, (b) follows from (a) and Proposition 3.29. □

5.6. Differentials of the gluing maps and final arguments. We summarise the results of the preceding sections and record our bounds for the differentials of the approximate gluing maps 𝒥 and 𝒥̂. The estimates for D𝒥̂ then give bounds for the diagonal (and so all) components of the L^{2} metric g and completes the proof of Theorem 1.1. Combining these metric bounds with results of Donaldson in [D89] then completes the proof of Theorem 1.2. The following two theorems summarise the estimates obtained in §§5.3 to 5.5, the first following from Corollaries 5.22, 5.29, and 5.32 and the second from Theorems 5.23, 5.30, and 5.33.

Theorem 5.34. Let 𝒥 : 𝒯/Γ → M_{X,k}^{∗}(g) be a gluing map and assume that the conditions of Theorem 5.8 hold. Then for sufficiently small λ_{0} > 0 and any t ∈ 𝒯, there exists a constant C = C(g_{0}, 𝒯) such that the following bounds hold:

(a) ∥D𝒥(∂/∂t_{I}^{α})∥_{L^{2}(X,g)} ≤ C,

(b) ∥D𝒥(∂/∂ρ_{I}^{β})∥_{L^{2}(X,g)} ≤ C\overline{λ}^{1/2},

(c) ∥D𝒥(∂/∂x_{I}^{μ})∥_{L^{2}(X,g)} ≤ C(1 + \overline{λ}^{1/2+δ}λ_{I}^{−δ}),

(d) ∥D𝒥(∂/∂λ_{I})∥_{L^{2}(X,g)} ≤ C(1 + \overline{λ}^{1/2+δ}λ_{I}^{−1/2−δ}).

Theorem 5.35. Let 𝒥̂ : 𝒯/Γ → M_{X_{0},k}^{∗}(g_{0}) be a gluing map and assume that the conditions of Theorem 5.8 hold. Then for any 0 < δ < 1/2, sufficiently small λ_{0} > 0 and any t ∈ 𝒯, there exists a constant C = C(δ, g_{0}, 𝒯) such that the following bounds hold:

(a) ∥D𝒥̂(∂/∂t_{I}^{α})∥_{L^{2}(X_{0},g_{0})} ≤ C,

(b) ∥D𝒥̂(∂/∂ρ_{I}^{β})∥_{L^{2}(X_{0},g_{0})} ≤ C\overline{λ}^{1/2},

(c) ∥D𝒥̂(∂/∂x_{I}^{μ})∥_{L^{2}(X_{0},g_{0})} ≤ C(1 + \overline{λ}^{1/2+δ}λ_{I}^{−δ}),

(d) ∥D𝒥̂(∂/∂λ_{I})∥_{L^{2}(X_{0},g_{0})} ≤ C(1 + \overline{λ}^{1/2+δ}λ_{I}^{−1/2−δ}).

It remains to reinterpret the bounds of Theorem 5.35 in terms of the corresponding bounds for the diagonal components of the L^{2} metric g.

Corollary 5.36. Under the hypotheses of Theorem 5.35, the following bounds hold:

(a) g(∂/∂t_{I}^{α}, ∂/∂t_{I}^{α}) ≤ C,

(b) g(∂/∂ρ_{I}^{β}, ∂/∂ρ_{I}^{β}) ≤ C\overline{λ},

(c) g(∂/∂x_{I}^{μ}, ∂/∂x_{I}^{μ}) ≤ C(1 + \overline{λ}^{1+2δ}λ_{I}^{−2δ}),

(d) g(∂/∂λ_{I}, ∂/∂λ_{I}) ≤ C(1 + \overline{λ}^{1+2δ}λ_{I}^{−1−2δ}).

Recall that the g-length of a path (s_{0}, s_{1}) ∋ s → A(s) ∈ M_{X_{0},k}^{∗}(g_{0}) is computed by

∫_{s_{0}}^{s_{1}} \sqrt{g} (\frac{∂A}{∂s}, \frac{∂A}{∂s}) ds ≤ ∫_{s_{0}}^{s_{1}} ∥\frac{∂A}{∂s}∥_{L^{2}(X_{0},g_{0})} ds.

The proofs of our main results are now essentially complete.

Proof of Theorem 1.1: Since 0 < δ < 1/2, the bounds of Theorem 5.36 imply that the gluing neighbourhoods 𝒱 = 𝒥̂(𝒯^{0}/Γ) have finite g-volume and g-diameter. Therefore, the bubbling ends of M_{X_{0},k}^{∗}(g_{0}) have finite g-volume and g-diameter since the entire moduli space is covered by finitely many such neighbourhoods. Away from the Uhlenbeck boundary, gluing neighbourhoods consist simply of C^{∞} Kuranishi charts. The conical ends corresponding to Kuranishi charts around the reducible connections have finite g-volume and g-diameter by Theorem 1 [G-P89]. □

Next we consider the relationship between the metric completion and the Uhlenbeck compactification of the anti-self-dual moduli space. Let d_{2} be the distance function on M_{X_{0},k}^{∗}(g_{0}) defined by the L^{2} metric g. Thus, if [A], [B] are two points in M_{X_{0},k}^{∗}(g_{0}), then d_{2}([A], [B]) is the infimum over all g-lengths of paths in M_{X_{0},k}^{∗}(g_{0}) joining [A], [B]. If the two points lie in different path components of the moduli space, then set d_{2}([A], [B]) = ∞. Since b^{+}(X_{0}) = 0, the moduli space has at most finitely many path components; we say that M_{X_{0},k}^{∗}(g_{0}) has finite g-diameter if the sum of the g-diameters of the connected components is finite. In [D89], Donaldson constructs two other distance functions, D_{2} and D_{2}^{ε}, for any fixed ε > 0. First, given points [A], [B] in B_{X_{0},k}^{∗}, set

D_{2}([A], [B]) ≡ inf_{u∈G} ∥A − u^{∗}B∥_{L^{2}(X_{0},g_{0})}.

Lemma 2 [D89] (or Lemma 4.2.4 [D-K]) show that D_{2} is a well-defined distance function on B_{X_{0},k}^{∗}. Moreover, Lemma 1 [D89] shows that D_{2}([A], [B]) is equal to the distance function defined in the usual way by the L^{2} metric on B_{X_{0},k}^{∗} as the infimum over g-lengths of paths in B_{X_{0},k}^{∗} joining [A] and [B]. One then obtains a second distance function on M_{X_{0},k}^{∗}(g_{0}) by restriction. Define an ε-neighbourhood of M_{X_{0},k}^{∗}(g_{0}) in B_{X_{0},k}^{∗} by

B_{X_{0},k}^{∗,ε} ≡ {[A] ∈ B_{X_{0},k}^{∗} : ∥F_{A}^{+,g_{0}}∥_{L^{2}(X_{0},g_{0})} < ε}.

Then D_{2}^{ε}([A], [B]) is defined as infimum of the g-lengths of paths in B_{X_{0},k}^{∗,ε} joining two points [A] and [B] in B_{X_{0},k}^{∗,ε}. One now obtains a third distance function on M_{X_{0},k}^{∗}(g_{0}) by restriction. The three distance functions d_{2}, D_{2}, and D_{2}^{ε} on M_{X_{0},k}^{∗}(g_{0}) are related by

(5.31)  D_{2}([A], [B]) ≤ D_{2}^{ε}([A], [B]) ≤ d_{2}([A], [B]),  [A], [B] ∈ M_{X_{0},k}^{∗}(g_{0}).

To show that the d_{2}-completion of M_{X_{0},k}^{∗}(g_{0}) is homeomorphic to the Uhlenbeck compactification \overline{M}_{X_{0},k}^{u}(g_{0}), it is enough to show that a sequence [A^{α}] in M_{X_{0},k}^{∗}(g_{0}) is d_{2}-Cauchy if and only if it is convergent in the Uhlenbeck topology. For the metric D_{2}^{ε}, one has

Theorem 5.37. [D89, Theorem 4] For any ε > 0, the D_{2}^{ε}-completion of M_{X_{0},k}^{∗}(g_{0}) is homeomorphic to \overline{M}_{X_{0},k}^{u}(g_{0}). □

Thus Donaldson’s result gives part of the proof of Theorem 1.2: Suppose a sequence [A^{α}] in M_{X_{0},k}^{∗}(g_{0}) is d_{2}-Cauchy. According to Eq. (5.31), it must also be D_{2}^{ε}-Cauchy and so is convergent in the Uhlenbeck topology by Theorem 5.37 or simply by Proposition 6 [D89]. The proof of the reverse direction, namely that a sequence [A^{α}] which is convergent in the Uhlenbeck toplogy is also d_{2}-Cauchy, is included in [F94].

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Department of Mathematics, Harvard University, Cambridge, MA 02138. E-mail: feehan@math.harvard.edu.