PACIFIC JOURNAL OF MATHEMATICS Vol. 200, No. 1, 2001

CRITICAL-EXPONENT SOBOLEV NORMS AND THE SLICE THEOREM FOR THE QUOTIENT SPACE OF

CONNECTIONS

Paul M.N. Feehan

Following Taubes, we describe a collection of critical-expo-nent Sobolev norms, discuss their embedding and multiplication properties, and describe optimal Green’s operator estimates where the constants depend at most on the first positive eigenvalue of the covariant Laplacian of a G connection and the L^{2} norm of the connection’s curvature, for arbitrary compact Lie groups G. Using these critical-exponent norms, we prove a sharp, global analogue of Uhlenbeck’s Coulomb gaugefixing theorem, where the usual product connection over a ball is replaced by an arbitrary reference connection over the entire manifold. We also prove a quantitative version of the conventional slice theorem for the quotient space of G connections, with an invariant and sharp characterization of those points in the quotient space which are contained in the image of an L^{4} ball in the Coulomb-gauge slice. Our gauge-fixing and slice theorems use L_{1}^{2} distance functions on the quotient space and the estimate constants depend at most on the first positive eigenvalue of the covariant Laplacian of the reference connection and the L^{2} norm of its curvature.

1. Introduction.

The use of certain “critical-exponent” Sobolev norms is an important feature of methods employed by Taubes to solve the anti-self-dual and related non-linear elliptic partial differential equations [23], [24], [25]. Indeed, the estimates one can obtain using these critical-exponent norms appear to be the best possible when one needs to bound the norm of a Green’s operator for a Laplacian, depending on a connection varying in a non-compact family, in terms of minimal data such as the first positive eigenvalue of the Laplacian or the L^{2} norm of the curvature of the connection. Despite their utility, particularly in applications where an optimal analysis is required for gluing or degeneration problems (for example, when considering Uhlenbeck-bubbling families of anti-self-dual connections or PU(2) monopoles), these methods are not widely known. Following Taubes [21], [23], [24], [25] we describe a collection of critical-exponent Sobolev norms and general Green’s operator estimates depending only on first positive eigenvalues or the L^{2} norm of the connection’s curvature. These estimates are especially useful both for the construction of gluing maps, in the case of either anti-self-dual connections [24] or, more recently, in the case of PU(2) monopoles [6], [7], [9] and for analyzing their asymptotic behavior with respect to Uhlenbeck limits of the underlying gluing data. We apply them here to prove an optimal slice theorem for the quotient space of connections. The result is “optimal” in the sense that if a point [A] in the quotient space is known to be just L_{1}^{2}-close enough to a reference point [A_{0}] (see below for the precise statement), then A can be placed in Coulomb gauge relative to A_{0}, with all constants depending at most on the first positive eigenvalue of the covariant Laplacian defined by A_{0} and the L^{2} norm of the curvature of A_{0}. Such slice theorems are particularly advantageous when analyzing gluing maps and their differentials in situations (such as those of [8], [9]) where the underlying gluing data is allowed to “bubble”. In this paper we shall for simplicity only consider connections over four-dimensional manifolds, but the methods and results can adapted to the case of manifolds of arbitrary dimension, as in [26], to prove slice theorems applicable to cases where the reference connection is allowed to degenerate.

1.1. Critical-exponent Sobolev norms and the slice theorem. Suppose that X is a closed, Riemannian four-manifold, that G is a compact Lie group, and that B_{E}^{k,p} = A_{E}^{k,p}/G_{E}^{k+1,p} is the quotient space of L_{k}^{p} connections on a G bundle E modulo the Banach Lie group of L_{k+1}^{p} gauge transformations. Here, the integer k ≥ 1 and the Sobolev exponent 1 < p < ∞ obey the constraint (k + 1)p > 4, so L_{k+1}^{p}(X) ⊂ C^{0}(X) and gauge transformations in G_{E}^{k+1,p} are continuous. When (k + 1)p = 4 we have the “borderline”, “critical”, or “limiting case” of the Sobolev embedding theorem: L_{k+1}^{p}(X) ⊂ L^{q}(X) for all q < ∞ but not q = ∞.

A connection A ∈ A_{E}^{k,p} is in Coulomb gauge relative to a reference connection A_{0} if d_{A_{0}}^{∗}(A − A_{0}) = 0 and it is a standard result that S_{A_{0}} = A_{0} + Ker d_{A_{0}}^{∗} ⊂ A_{E}^{k,p} provides a slice for the action of the gauge group G_{E}^{k+1,p} [2], [5], [10], [11], [13], [14], [16], [18]. (See Proposition 3.4 for a detailed statement.) More exactly, if B_{A_{0}}^{k,p}(ε) is the L_{k}^{p} ball in S_{A_{0}} with center A_{0} and L_{k,A_{0}}^{p}-radius ε and Stab_{A_{0}} ⊂ G_{E}^{k+1,p} is the stabilizer of A_{0}, then the projection π : B_{A_{0}}^{k,p}(ε)/ Stab_{A_{0}} → B_{E}^{k,p} is a homeomorphism onto its image and thus contains a small enough L_{k}^{p} ball

B_{[A_{0}]}^{k,p}(η) = {[A] ∈ B_{E}^{k} : dist_{L_{k,A_{0}}^{p}} ([A], [A_{0}]) < η} , (1.1) where gauge-invariant distance functions on the G_{E}^{k+1,p}-quotient are defined by

dist_{L_{k′,A_{0}}^{p′}} ([A], [A_{0}]) = inf_{u∈G_{E}^{k+1,p}} ∥u(A) − A_{0}∥_{L_{k′+1,A_{0}}^{p′}} , (1.2)

whenever L_{k}^{p} ⊂ L_{k′}^{p′}. One unsatisfactory aspect of the standard slice theorem concerns the dependence of the constants ε([A_{0}], k, p) and η([A_{0}], k, p) above on the orbit [A_{0}]—in particular on the curvature F_{A_{0}}—when k and p are large enough that gauge transformations in G_{E}^{k+1,p} are continuous. Even in the minimal cases, k = 1 and p > 2 or k = 2 and p = 2, the constants ε, η depend unfavorably on [A_{0}] when the curvature F_{A_{0}} bubbles. This makes it difficult to analyze the asymptotic behavior of Taubes’ gluing maps [20], [22], [23], [24] and their differentials on neighborhoods of points in the Uhlenbeck boundary of the moduli space of anti-self-dual connections, since the balls B_{A_{0}}^{k,p}(ε) and B_{[A_{0}]}^{k,p}(η) tend to shrink as [A_{0}] approaches an Uhlenbeckboundary point. For example, if the connection A_{0} is anti-self-dual, then its energy is bounded by a constant depending only on the topology of E via the Chern-Weil identity [5, §2.1.4], whereas ∥F_{A_{0}}∥_{L^{p}} (with p > 2) or ∥F_{A_{0}}∥_{L_{1,A_{0}}^{2}} tends to infinity as the curvature of A_{0} bubbles.

Our main purpose in this article is to prove a global analogue, Theorem 1.1, of Uhlenbeck’s local Coulomb gauge-fixing theorem [26, Theorems 1.3 & 2.1] and a corresponding slice theorem, Theorem 1.2, where the radii of the coordinate balls on the quotient B_{E}^{k,p} depend only on ∥F_{A_{0}}∥_{L^{2}} and the least positive eigenvalue ν_{0}[A_{0}] of the Laplacian d_{A_{0}}^{∗}d_{A_{0}} on Ω^{0}(𝔤_{E}). The key difficulty in establishing Theorem 1.1 is to ensure that the constants depend at most on ∥F_{A_{0}}∥_{L^{2}} and ν_{0}[A_{0}]: To guarantee this minimal dependence, we employ critical-exponent Sobolev norms (defined below) to circumvent the fact that when (k + 1)p = 4 the standard Sobolev embedding and multiplication theorems fall just short of what one needs to give the quotient B_{E}^{k,p} = A_{E}^{k,p}/G_{E}^{k+1,p} a manifold structure (see Section 4). Such norms were introduced by Taubes for related purposes in [23].

1.2. Statement of results. For clarity, we now fix p = 2 and k ≥ 2 and define the following distance functions on the quotient space B_{E}^{k} = A_{E}^{k}/G_{E}^{k+1} of L_{k}^{2} connections modulo L_{k+1}^{2} gauge transformations,

dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) = inf_{u∈G_{E}^{k+1}} (∥u(A) − A_{0}∥_{L^{2♯,4}} + ∥d_{A_{0}}^{∗}(u(A) − A_{0})∥_{L^{♯,2}}) ,
dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) = inf_{u∈G_{E}^{k+1}} (∥u(A) − A_{0}∥_{L_{1,A_{0}}^{2}} + ∥d_{A_{0}}^{∗}(u(A) − A_{0})∥_{L^{♯,2}}) ,

(1.3) where the norms of a ∈ Ω^{1}(𝔤_{E}) are defined by (see Equations (4.1), (4.2), and (4.3)),

∥a∥_{L^{♯}(X)} = sup_{x∈X} ∥ dist^{−2}(x, ·)|a|∥_{L^{1}(X)},
∥a∥_{L^{2♯}(X)} = sup_{x∈X} ∥ dist^{−1}(x, ·)|a|∥_{L^{2}(X)},
∥a∥_{L^{♯,2}(X)} = ∥a∥_{L^{2}(X)} + ∥a∥_{L^{♯}(X)},
∥a∥_{L^{2♯,4}(X)} = ∥a∥_{L^{4}(X)} + ∥a∥_{L^{2♯}(X)},
∥a∥_{L_{1,A_{0}}^{2}(X)} = (∥a∥_{L^{2}(X)}^{2} + ∥∇_{A_{0}}a∥_{L^{2}(X)}^{2})^{1/2} .

Here, dist(x, y) denotes the geodesic distance between points x, y ∈ X. The distance function dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) is bounded by scale invariant norms,

∥a∥_{L^{4}(X,g)} +∥∇_{A}^{g}a∥_{L^{2}(X,g)} + sup_{x∈X} ∥ dist_{g}^{−2}(x, ·)|d_{A_{0}}^{∗_{g}}a|∥_{L^{1}(X,g)}, a ∈ Ω^{1}(𝔤_{E}),

since the L^{4/ℓ} norm on ⊗^{ℓ}(T^{∗}X) is conformally invariant, while the third term is invariant under constant rescalings g ↦ g̃ = λ^{−2}g of the metric, as

d_{A_{0}}^{∗g̃}a = λ^{2}d_{A_{0}}^{∗_{g}}a, dist−2 ; g̃(x, y) = λ^{2} dist_{g}^{−2}(x, y) and dVg̃ = λ^{−4}dV_{g}. Similarly

for dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]). Like the L^{4} norm, the L^{2♯} norm on one-forms is scale-

invariant. Our first result is the following global analogue of Uhlenbeck’s theorem and complements results of Taubes in [23, §6]:

Theorem 1.1. Let X be a closed, smooth four-manifold with metric g and let G be a compact Lie group. Then there are positive constants c, z with the following significance. Let E be a G bundle over X and suppose that k ≥ 2 is an integer. Given a point [A_{0}] in B_{E}^{k}, let ν_{0}[A_{0}] be the least positive eigenvalue of the Laplacian d_{A_{0}}^{∗}d_{A_{0}} on Ω^{0}(𝔤_{E}) and set K_{0} = (1 + ν_{0}[A_{0}]^{−1})(1 + ∥F_{A_{0}}∥_{L^{2}}). Let ε_{1} be a constant satisfying 0 < ε_{1} ≤ zK_{0}^{−2}(1 + ν_{0}[A_{0}]^{−1/2})^{−1}. Then:

(1) For any [A] ∈ B_{E}^{k} with dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) < ε_{1}, there is a gauge

transformation u ∈ G_{E}^{k+1}, unique up to an element of the stabilizer Stab_{A_{0}} ⊂ G_{E}^{k+1}, such that:

(a) d_{A_{0}}^{∗}(u(A) − A_{0}) = 0,

(b) ∥u(A) − A_{0}∥_{L^{2♯,4}} ≤ cK_{0} dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]).

(2) For any [A] ∈ B_{E}^{k} with dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) < ε_{1}, there is a gauge

transformation u ∈ G_{E}^{k+1}, unique up to an element of the stabilizer Stab_{A_{0}} ⊂ G_{E}^{k+1}, such that:

(a) d_{A_{0}}^{∗}(u(A) − A_{0}) = 0,

(b) ∥u(A) − A_{0}∥_{L^{2♯,4}} ≤ cK_{0} dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]),

(c) ∥u(A) − A_{0}∥_{L_{1,A_{0}}^{2}} ≤ cK_{0} dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]).

In Theorem 2.1 of [26] the L^{2} norm of the curvature F_{A} of a local connection matrix A over the unit ball in ℝ^{4} provides a natural (gauge-invariant) measure of the distance from [A] to [Γ], where Γ is the product connection. Uhlenbeck’s theorem guarantees the existence of an L_{k+1}^{p} gauge transformation u taking an L_{k}^{p} connection A on the product bundle over the unit four-ball, with product connection Γ, to a connection u(A) satisfying d_{Γ}^{∗}(u(A) − Γ) = 0 and ∥u(A) − Γ∥_{L_{1}^{2}} ≤ c∥F_{A}∥_{L^{2}}; one only requires that ∥F_{A}∥_{L^{2}} be smaller than a universal constant.

We next have the following refinement of the standard slice theorem for the quotient space B_{E}^{k}. The observation that an L^{4}-ball in Ker d_{A_{0}}^{∗} provides a slice for G_{E}^{k+1} was pointed out to us Mrowka; that slightly smaller L^{2♯,4} and L_{1,A_{0}}^{2} balls provide slices follows from the second of our two proofs of Theorem 1.1 in Section 8. For any ε > 0, we define open balls

B_{[A_{0}]}^{1,♯,2}(ε) = {[A] ∈ B_{E}^{k} : dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) < ε} ⊂ B_{E}^{k},
B_{[A_{0}]}^{1,∗,2}(ε) = {[A] ∈ B_{E}^{k} : dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) < ε} ⊂ B_{E}^{k},
B_{A_{0}}^{4}(ε) = {A ∈ A_{E}^{k} : d_{A_{0}}^{∗}(A − A_{0}) = 0 and ∥A − A_{0}∥_{L^{4}(X)} < ε}
⊂ S_{A_{0}},

(1.4)

where S_{A_{0}} = {A_{0}} + Ker(d_{A_{0}}^{∗}|_{L_{k}^{2}}) ⊂ A_{E}^{k} is the slice through A_{0}. We let

B̄_{[A_{0}]}^{1,♯,2}(ε) and B_{[A_{0}]}^{1,∗,2}(ε) denote the closed balls.

Theorem 1.2. Let X be a closed, smooth four-manifold with metric g and let G be a compact Lie group. Then there are positive constants c_{1}, c_{2}, z with the following significance. Let E be a G bundle over X, let k ≥ 2 be an integer, and suppose that [A_{0}] ∈ B_{E}^{k}. Then:

(1) For ε_{0} such that 0 < ε_{0} < z(1 + ν_{0}[A_{0}]^{−1/2})^{−1}, the projection π : B_{A_{0}}^{4}(ε_{0})/ Stab_{A_{0}} → B_{E}^{k}, A ↦ [A], is a homeomorphism onto an open neighborhood of [A_{0}] ∈ B_{E}^{k} and a diffeomorphism on the open subset where Stab_{A_{0}} / Center(G) acts freely;

(2) For any constant ε_{1} satisfying 0 < ε_{1} ≤ zK_{0}^{−2}(1 + ν_{0}[A_{0}]^{−1/2})^{−1} we have the following inclusions of open neighborhoods in B_{E}^{k}:

B_{[A_{0}]}^{1,♯,2}(ε_{1}) ⊂ B_{[A_{0}]}^{1,∗,2}(c_{1}ε_{1}) ⊂ π(B_{A_{0}}^{4}(c_{2}K_{0}ε_{1})).

That sharper versions of the standard slice theorem (as in [5], [10], [11], for example) would hold is suggested by related results of Taubes, namely [21, Lemma A.1] and [23, Lemma 6.5]: For example, they show that if u is an L_{2}^{2} gauge transformation intertwining L_{1}^{2} connections A_{i}, i = 1, 2, obeying a slice condition d_{A_{0}}^{∗}(A_{i} − A_{0}) = 0 defined by an L_{1}^{2} connection A_{0}, then u is necessarily in C^{0}. Moreover, transition functions relating neighborhoods of the origin in Ker(d_{A_{0}}^{∗}|_{L_{1}^{2}}) and Ker(d_{A_{0}+a}^{∗}|_{L_{1}^{2}}), where a is L_{1,A_{0}}^{2}-small, are constructed in [23, Lemma 6.5]; the constants depend only on ∥F_{A_{0}}∥_{L^{2}} and ν_{0}[A_{0}]. (See [23, §6] for detailed statements and related results.) The proof of Theorem 1.1 makes use of methods developed in [21], [23], [24]. To illustrate applications of the methods of Sections 4 and 5 and to point to possible generalizations of the estimates in this article, we derive some elliptic estimates for the linearization of the anti-self-dual equation in Section 5.2.

1.3. Outline of the proofs. Assertion (1) of Theorem 1.2 is proved in Section 3. The proof that the projection map π : B_{A_{0}}^{4}(ε_{0}) → B_{E}^{k} is a local diffeomorphism away from connections with non-minimal stabilizer essentially follows Uhlenbeck’s verification of “openness” in her proof of Theorem 2.1 in [26] via the method of continuity (see Lemma 3.6). The proof that the L^{4} ball B_{A_{0}}^{4}(ε_{0}) injects into the quotient (see Lemma 3.7) was suggested to us by Mrowka. The remainder of our article is taken up with the proof of Theorem 1.1 and hence Assertion (2) of Theorem 1.2.

In Section 4 we introduce the family of critical-exponent Sobolev norms, L_{k,A_{0}}^{♯,2}, k = 0, 1, 2, used to complete the proof of Theorem 1.1 and in Section 5 we describe the crucial embedding theorems enjoyed by those Sobolev spaces, as well as estimates for the Green’s operator of the Laplacian d_{A_{0}}^{∗}d_{A_{0}}. In particular, L_{k,A_{0}}^{p} ⊂ L_{k,A_{0}}^{♯,2}, for every p > 2 while, in the other direction, L_{2,A_{0}}^{♯,2} ⊂ C^{0}. The latter embedding is the key motivation for the definition of these norms and it greatly facilitates the derivation of Green’s operator estimates, in a wide number of applications in gauge theory [23], [24], with minimal dependence on the curvature of the connection A_{0}. The main ideas and embedding results in Sections 4 and 5 are due to Taubes [21], [23], [24], [25], so these sections are essentially expository. An earlier exposition from a somewhat different perspective, due to Donaldson, of Taubes’ methods and some applications appears in [4]. The estimates of Section 5 are stated only in the four-dimensional case. While we might expect all of them to hold, in some form, for higher dimensions we confine our attention to dimension four as our intended applications are primarily concerned with smooth fourmanifold topology. In essence, the critical-exponent norms make a virtue out of necessity of the familiar fact that while the Green’s operator of the Laplacian d^{∗}d on C^{∞}(X) maps L^{p}(X) into L^{2p/(2−p)}(X) for 1 < p < 2, it does not map L^{2}(X) into L^{∞}(X) [19, Chapter V]. We recall that an Orlicz space L_{φ} can be used to provide the “best target space” for an embedding of L_{2}^{2}(X) [1, Chapter 8]. Here, we may instead view L_{2}^{♯,2}(X) as providing the “best domain space” for an embedding into L^{∞}(X), since L_{2}^{p}(X) ⊂ L_{2}^{♯,2}(X) ⊂ L^{∞}(X) for all p > 2.

We give two proofs of Theorem 1.1. For our first proof, in Section 6, we essentially follow the strategy of Uhlenbeck [26] and apply the method of continuity. The difficult step here (in establishing openness—see Section 6.3) is to prove that the intrinsic, gauge-invariant L_{1,A_{0}}^{♯,2} and L_{1,A_{0}}^{♯,2} distances in the quotient B_{E}^{k} bound the L^{2♯,4} and L_{1,A_{0}}^{2} norms in the slice S_{A_{0}} ⊂ A_{E}^{k}: This is the point in our first proof where we use the critical-exponent estimates derived in Section 5 to control gauge transformations. The proof of “closedness” uses a compactness argument and is given in Section 6.2. The proofs of Theorems 1.1 and 1.2 are completed in Section 6.3.

Our second proof of Theorem 1.1 occupies Sections 7 and 8. In Section 7 we show that the exponential map Exp : Ω^{0}(𝔤_{E}) → G_{E} extends to a continuous map Exp : L_{2}^{♯,2}(𝔤_{E}) → L_{2}^{♯,2}(𝔤_{E}) and that the resulting space of L_{2}^{♯,2}-gauge transformations G_{E}^{2,♯,2} is a Banach Lie group. In particular, L_{2}^{♯,2}-gauge transformations are continuous and are contained in G_{E}^{2,p} for every p > 2. The Sobolev multiplication and composition results for the critical-exponent norms then allow us to apply the inverse function theorem directly in Section 8, while still ensuring that all constants depend at most on ν_{0}[A_{0}] and ∥F_{A_{0}}∥_{L^{2}}. We first use the compactness result of Section 6.1 to establish the existence of gauge transformations w in G_{E}^{3} which minimize the L_{1,A_{0}}^{♯,2} and L_{1,A_{0}}^{♯,2} distances in the quotient B_{E}^{k}. Then, assuming the norm ∥w(A) − A_{0}∥_{L^{2♯,4}} or ∥w(A) − A_{0}∥_{L_{1,A_{0}}^{2}} is sufficiently small, we use the Sobolev embedding and multiplication theorems of Sections 4, 5, and 7 and a quantitative version of the inverse function theorem to prove the existence of a gauge transformation v ∈ G_{E}^{3} such that d_{A_{0}}^{∗}(u(A) − A_{0}) = 0, u = vw ∈ G_{E}^{k+1}, and ∥u(A) − A_{0}∥_{L^{2♯,4}} and ∥u(A) − A_{0}∥_{L_{1,A_{0}}^{2}} are controlled by dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) and dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]), respectively.

Acknowledgments. We warmly thank Tom Mrowka for helpful and stimulating discussions on gauge theory and his suggestion that L^{4}-balls inject into the quotient, and warmly thank Cliff Taubes for very helpful discussions and especially for explaining his ideas and methods in [21], [23]. We would also like to thank the Harvard Mathematics Department, where this article was completed in September 1996, and the National Science Foundation for their generous hospitality and support during the preparation of this article.

2. Preliminaries.

We assume throughout this article that X is a closed, connected, smooth, four-manifold with Riemannian metric g. Let G be a compact Lie group with matrix representation ρ : G ⊂ SO(𝔼) = SO(r) where 𝔼 ≃ ℝ^{r} as a real inner product space, let P be a principal G bundle, and let E = P ×_{ρ} 𝔼 be the corresponding Riemannian vector bundle associated to P by the representation ρ. Let 𝔤_{E} ⊂ 𝔤𝔩(E) be the bundle of Lie algebras associated to P via the adjoint representation Ad : G → Aut(𝔤) of G on its Lie algebra 𝔤 and viewed as a subbundle of 𝔤𝔩(E) via the induced representation ρ_{∗} : 𝔤 ⊂ 𝔰𝔬(𝔼).

Given the covariant derivative ∇_{A} : C^{∞}(E) → C^{∞}(T^{∗}X ⊗ E), we define the exterior covariant derivative d_{A} : Ω^{i}(E) → Ω^{i+1}(E) in the usual way by setting d_{A} = ∇_{A} on Ω^{0}(E) = C^{∞}(E) and extending d_{A} to Ω^{i}(E) = C^{∞}(Λ^{i} ⊗ E), where Λ^{i} := Λ^{i}(T^{∗}X), according to the rule d_{A}(ω ∧ v) = dω ∧ v + (−1)^{i}ω ∧ d_{A}v for ω ∈ Ω^{i}(X) and v ∈ Ω^{j}(E).

For any integer k ≥ 0, exponent 1 ≤ p ≤ ∞, and L_{k}^{p} connection A_{0} on E we define the L_{k}^{p} Sobolev completion, L_{k}^{p}(Λ^{ℓ} ⊗ E), of Ω^{ℓ}(E) with respect to the norm

∥s∥_{L_{k,A_{0}}^{p}(X)} := (∑_{j=0}^{k} ∥∇_{A_{0}}^{j}s∥_{L^{p}(X)}^{p})^{1/p} .

We define the action of a C^{∞} gauge transformation u ∈ G_{E} on a C^{∞} connection A on the bundle E by pushforward, so u(A) := A − (d_{A}u)u^{−1}. Fix a connection A_{0} ∈ A_{E}, let A_{E}^{k} = A_{0} + L_{k}^{2}(Λ^{1} ⊗ 𝔤_{E}), and define

G_{E}^{k+1} := {u ∈ L_{k+1}^{2}(𝔤𝔩(E)) : u ∈ G a.e.} ⊂ L_{k+1}^{2}(𝔤𝔩(E)).

The space G_{E}^{k+1} is a Banach Lie group, with Lie algebra T_{id_{E}}G_{E}^{k+1}=L_{k+1}^{2}(𝔤_{E}), and acts smoothly on A_{E}^{k} with quotient B_{E}^{k} := A_{E}^{k}/G_{E}^{k+1} endowed with the quotient L_{k}^{2} topology.

The stabilizer subgroup Stab_{A} ⊂ G_{E}^{k+1} for a connection A on E always contains the center Center(G) ⊂ G. We let A_{E}^{∗,k} ⊂ A_{E}^{k} denote the space of connections A ∈ A_{E}^{k} with minimal stabilizer Stab_{A} = Center(G) and let B_{E}^{∗,k} = A_{E}^{∗,k}/G_{E}^{k+1}. As usual, the stabilizer subgroup Stab_{A} ⊂ G_{E} can be identified with a closed subgroup of G ⊂ GL(E|_{x_{0}}) for any point x_{0} ∈ X by parallel translation with respect to the connection A. Let 𝔰𝔱𝔞𝔟_{A} denote the Lie algebra of Stab_{A}, so 𝔰𝔱𝔞𝔟_{A} = Ker{d_{A} : L_{k+1}^{2}(𝔤_{E}) → L_{k}^{2}(Λ^{1} ⊗ 𝔤_{E})}.

Throughout the article, we use c or z to denote positive constants which depend at most on the Riemannian manifold (X, g) and the group G; constants may increase from one line to the next and are not renamed unless clarity demands otherwise.

3. The slice theorem.

In this section we prove the first assertion of Theorem 1.2—see Proposition 3.4 below—namely, that a small enough L^{4}-ball B_{A_{0}}^{4}(ε_{0})/ Stab_{A_{0}} provides a slice for the action of G_{E}^{k+1}. The proof that the projection π :

B_{A_{0}}^{4}(ε_{0})/ Stab_{A_{0}} → B_{E}^{k} is injective (Lemma 3.7) was suggested to us by Mrowka.

Let k ≥ 2 be an integer. The Banach Lie group G_{E}^{k+1} has Lie algebra T_{id_{E}}G_{E}^{k+1} = L_{k+1}^{2}(𝔤_{E}) and exponential map Exp : L_{k+1}^{2}(𝔤_{E}) → G_{E}^{k+1} given by ζ ↦ u = Exp ζ. Recall that Stab_{A} = {γ ∈ G_{E}^{k+1} : γ(A) = A} may be identified with a Lie subgroup of G and has Lie algebra 𝔰𝔱𝔞𝔟_{A} = Ker(d_{A}|_{L_{k+1}^{2}}). The operator d_{A}^{∗} : L_{k+2}^{2}(Λ^{1} ⊗ 𝔤_{E}) → L_{k+1}^{2}(𝔤_{E}) has closed range and we have an L^{2}-orthogonal decomposition

(3.1)  T_{id_{E}}G_{E}^{k+1} = L_{k+1}^{2}(𝔤_{E})
= (Ker(d_{A}|_{L_{k+1}^{2}}))^{⊥} ⊕ Ker (d_{A}|_{L_{k+1}^{2}})
= Im (d_{A}^{∗}|_{L_{k+2}^{2}}) ⊕ Ker (d_{A}|_{L_{k+1}^{2}})
= (Ker (d_{A}|_{L_{k+1}^{2}}))^{⊥} ⊕ 𝔰𝔱𝔞𝔟_{A}.

Let Stab_{A}^{⊥} = L_{k+1}^{2} ∩ Stab_{A}^{⊥} = Exp((Ker d_{A}|_{L_{k+1}^{2}})^{⊥}), the second equality

following from the Sobolev composition lemma. The subspace Stab_{A}^{⊥} ⊂ G_{E}^{k+1} is closed and is a Banach submanifold of G_{E}^{k+1} with codimension dim 𝔰𝔱𝔞𝔟_{A}. From Claim 3.5 below we see that Stab_{A}^{⊥} is a slice near id_{E} ∈ G_{E}^{k+1} for the right action of Stab_{A} on G_{E}^{k+1}.

The map d_{A} : L_{k+1}^{2}(𝔤_{E}) → L_{k}^{2}(Λ^{1} ⊗ 𝔤_{E}) has closed range and so we have an L^{2}-orthogonal decomposition

(3.2)  T_{A}A_{E}^{k} = L_{k}^{2} (Λ^{1} ⊗ 𝔤_{E})
= Im (d_{A}|_{L_{k+1}^{2}}) ⊕ Ker (d_{A}^{∗}|_{L_{k}^{2}})
= Im (d_{A}|_{L_{k+1}^{2}}) ⊕ K_{A},

of the tangent space to the space of L_{k}^{2} connections at A, where K_{A} = Ker(d_{A}^{∗}|_{L_{k}^{2}}).

The slice S_{A} ⊂ A_{E}^{k} through a connection A is given by S_{A} = A + K_{A}. If π is the projection from A_{E}^{k} onto B_{E}^{k} = A_{E}^{k}/G_{E}^{k+1}, denoted by A ↦ [A], we let

B_{A}(ε) = {A_{1} ∈ S_{A} : ∥A_{1} − A∥_{L_{k;A}^{2}} < ε}
= A + {a ∈ K_{A} : ∥a∥_{L_{k;A}^{2}} < ε}

be the open L_{k}^{2}-ball in S_{A} with center A and L_{k,A}^{2}-radius ε. Similarly, we let

B_{A}^{4}(ε) = {A_{1} ∈ S_{A} : ∥A_{1} − A∥_{L^{4}} < ε}
= A + {a ∈ K_{A} : ∥a∥_{L^{4}} < ε}

be the open ball in S_{A} with center A and L^{4}-radius ε.

The proof that the quotient space A_{E}^{k} is Hausdorff makes use of the following well-known technical result [10, Proposition A.5]. Note that the space G_{E}^{2} is neither a Banach Lie group nor does it act smoothly on A_{E}^{k} for k ≥ 1.

Lemma 3.1. Let E be a Hermitian bundle over a Riemannian manifold X and let k ≥ 2 be an integer. Suppose {A_{α}} and {B_{α}} are sequences of L_{k}^{2} unitary connections on E and that {u_{α}} is a sequence in G_{E}^{2} such that u_{α}(A_{α}) = B_{α}. Then the following hold.

(1) The sequence {u_{α}} is in G_{E}^{k+1};

(2) If {A_{α}} and {B_{α}} converge in A_{E}^{k} to limits A_{∞}, B_{∞}, then there is a subsequence {α′} ⊂ {α} such that {u_{α′}} converges in G_{E}^{k+1} to u_{∞} and B_{∞} = u_{∞}(A_{∞}).

We shall need the following quantitative version of the inverse function theorem here and especially in Section 8:

Theorem 3.2. Let Φ : E → F be a C^{ℓ} map of Banach spaces, for some ℓ ≥ 1, such that the differential (DΦ)_{x_{0}} : E → F has a continuous inverse (DΦ)_{x_{0}}^{−1} : F → E satisfying

∥(DΦ)_{x_{0}}^{−1}∥ ≤ K and ∥(DΦ)_{x} − (DΦ)_{x_{0}}∥ ≤ \frac{1}{2}K^{−1}, if ∥x − x_{0}∥ ≤ δ,

for some positive constants K and δ. Then the following hold:

(1) The restriction of Φ to the ball U = B^{E}(x_{0}, δ) is injective and Φ(U) = V is an open set in F containing the ball B^{F}(Φ(x_{0}), δ/(2K));

(2) The inverse map Φ^{−1} : V → U is C^{ℓ};

(3) If x_{1}, x_{2} ∈ B^{E}(x_{0}, δ), then ∥x_{1} − x_{2}∥ ≤ 2K∥Φ(x_{1}) − Φ(x_{2})∥.

For quantitative comparisons in this section, the following elementary fact will suffice:

Lemma 3.3. Let E, F be Banach spaces and let T ∈ Hom(E, F) have a right (left ) inverse S. If T̃ ∈ Hom(E, F) satisfies ∥T̃ − T ∥ < ∥S∥^{−1}, then T̃ also has a right (left ) inverse.

Proof. If S ∈ Hom(F, E) is a right inverse for T, so T S = id_{F}, then ∥(T̃ − T)S∥ ≤ ∥T̃ − T ∥∥S∥ < 1 and id_{E} + (T̃ − T)S is an invertible element of the Banach algebra End(E). Define S̃ = S(1 + (T̃ − T)S)^{−1}, so S̃T = id_{E} and S̃ is a right inverse for T̃. Similarly for left inverses. □

This consequence of the usual characterization of invertible elements of a Banach algebra will be invoked in the proof of Lemma 3.6.

Proposition 3.4. Let X be a closed, Riemannian four-manifold. Then there is a positive constant z with the following significance. Let E be a G bundle over X. Suppose that k ≥ 2 is an integer. Given [A_{0}] in B_{E}^{k}, let ν_{0}[A_{0}] be the least positive eigenvalue of the Laplacian Δ_{A_{0}}^{0} and let ε_{0} be a constant satisfying 0 < ε_{0} < z(1 + ν_{0}[A_{0}]^{−1/2})^{−1}. Then:

(1) The space B_{E}^{k} is Hausdorff;

(2) The subspace B_{E}^{∗,k} ⊂ B_{E}^{k} is open and is a C^{∞} Banach manifold with local parameterizations given by π : B_{A_{0}}^{4}(ε_{0}) → B_{E}^{∗,k};

(3) The projection π : A_{E}^{∗,k} → B_{E}^{∗,k} is a C^{∞} principal G_{E}^{k+1}/ Center(G) bundle;

(4) The projection π : B_{A_{0}}^{4}(ε_{0})/ Stab_{A_{0}} → B_{E}^{k} is a homeomorphism onto an open neighborhood of [A_{0}] ∈ B_{E}^{k} and a diffeomorphism on the subset where Stab_{A_{0}} / Center(G) acts freely.

Proof. The stabilizer Stab_{A_{0}} acts freely on G_{E}^{k+1} and thus on the Banach manifold G_{E}^{k+1} × S_{A_{0}}^{k} by (u, A) ↦ γ · (u, A) = (uγ^{−1}, γ(A)) and so the quotient G_{E}^{k+1} ×_{Stab_{A_{0}}} S_{A_{0}} is again a Banach manifold. We define a smooth map

Ψ : G_{E}^{k+1} ×_{Stab_{A_{0}}} S_{A_{0}} → A_{E}^{k}, [u, A] ↦ u(A). (3.3)

Our main task is to show that the map Ψ is (i) a local diffeomorphism onto its image and (ii) injective upon restriction to a sufficiently small neighborhood G_{E}^{k+1} ×_{Stab_{A_{0}}} B_{A_{0}}^{4}(ε_{0}). Given δ_{0} > 0, let B_{id_{E}}(δ_{0}) be the ball {u ∈ G_{E}^{k+1} : ∥u − id_{E}∥_{L_{k+1;A_{0}}^{2}} < δ_{0}} and let B_{id_{E}}^{⊥}(δ_{0}) = B_{id_{E}}(δ_{0}) ∩ Stab_{A_{0}}^{⊥}.

Claim 3.5. For small enough δ = δ(A_{0}, k), the ball B_{id_{E}}(δ) is diffeomorphic to an open neighborhood in B_{id_{E}}^{⊥}(δ) × Stab_{A_{0}}, with inverse map given by (u_{0}, γ) ↦ u = u_{0}γ.

Proof. The differential of the multiplication map

Stab_{A_{0}}^{⊥} × Stab_{A_{0}} → G_{E}^{k+1}, (u_{0}, γ) ↦ u_{0}γ,

at (id_{E}, id_{E}) is given by

Ker (d_{A_{0}}|_{L_{k+1}^{2}})^{⊥} ⊕ 𝔰𝔱𝔞𝔟_{A_{0}} → L_{k+1}^{2}(𝔤_{E}) (ζ, χ) ↦ u_{0}ζγ + u_{0}γχ,

and so is just the identity map with respect to the L^{2}-orthogonal decomposition (3.1) of the range. Hence, the Banach space implicit function theorem implies that there is a diffeomorphism from an open neighborhood of (id_{E}, id_{E}) onto an open neighborhood of id_{E} ∈ G_{E}^{k+1}. For small enough δ, we may suppose that if u ∈ B_{id_{E}}(δ), then u can be written uniquely as u = u_{0}γ with u_{0} ∈ B_{id_{E}}^{⊥}(δ) and γ ∈ Stab_{A_{0}}. □

Lemma 3.6. For any 0 < ε_{0} < \frac{1}{2}(1 + ν_{0}[A_{0}]^{−1/2})^{−1}, the map Ψ is a local diffeomorphism from G_{E}^{k+1} ×_{Stab_{A_{0}}} B_{A_{0}}^{4}(ε_{0}) onto its image in A_{E}^{k}.

Proof. We first restrict the map Ψ to a neighborhood B_{id_{E}}(δ_{0}) ×_{Stab_{A_{0}}} S_{A_{0}}, which is diffeomorphic to the neighborhood B_{id_{E}}^{⊥}(δ) × S_{A_{0}} in Stab_{A_{0}}^{⊥} ×S_{A_{0}} by Claim 3.5. The differential of the induced map Ψ : Stab_{A_{0}}^{⊥} ×S_{A_{0}} → A_{E}^{k}, (u, A) ↦ u(A), (3.4) at (id_{E}, A) := (id_{E}, A_{0} + a_{0}) is given by

(DΨ)_{(id_{E},A)} :T_{id_{E}} Stab_{A_{0}}^{⊥} ⊕T_{A}S_{A_{0}} → T_{A}A_{E}^{k},
(ζ, a) ↦ −d_{A}ζ + a = −d_{A_{0}}ζ − [a_{0}, ζ] + a,

where we recall that T_{A}S_{A_{0}} = K_{A_{0}} = Ker(d_{A_{0}}^{∗}|_{L_{k}^{2}}) and

T_{id_{E}} Stab_{A_{0}}^{⊥} = (Ker (d_{A_{0}}|_{L_{k+1}^{2}}))^{⊥} = Im (d_{A_{0}}^{∗}|_{L_{k+2}^{2}}) .

Using the L^{2}-orthogonal decomposition (3.2) of the range we see that the map

−d_{A_{0}}⊕id_{E} : (Ker (d_{A_{0}}|_{L_{1}^{2}}))^{⊥}⊕Ker (d_{A_{0}}^{∗}|_{L_{1}^{2}}) → Im (d_{A_{0}}|_{L_{1}^{2}})⊕Ker (d_{A_{0}}^{∗}|_{L_{1}^{2}})

given by (ζ, b) ↦ −d_{A_{0}}ζ +b is a Hilbert space isomorphism. More explicitly, the operator

d_{A_{0}} : (Ker (d_{A_{0}}|_{L_{1}^{2}}))^{⊥} → Im(d_{A_{0}}|_{L_{1}^{2}}) = (Ker (d_{A_{0}}^{∗}|_{L^{2}}))^{⊥}

has a two-sided inverse

G_{A_{0}}^{0}d_{A_{0}}^{∗} : Im (d_{A_{0}}|_{L_{1}^{2}}) → (Ker (d_{A_{0}}|_{L_{1}^{2}}))^{⊥} ,

where G_{A_{0}}^{0} is the Green’s operator for the Laplacian Δ_{A_{0}}^{0} = d_{A_{0}}^{∗}d_{A_{0}}: Indeed, G_{A_{0}}^{0}d_{A_{0}}^{∗}d_{A_{0}} = G_{A_{0}}^{0}Δ_{A_{0}}^{0} is the L^{2}-orthogonal projection Π_{A_{0}}^{0} from L_{1}^{2}(Λ^{1}⊗𝔤_{E}) onto (Ker(d_{A_{0}}|_{L_{1}^{2}}))^{⊥} and d_{A_{0}}G_{A_{0}}^{0}d_{A_{0}}^{∗} is the L^{2}-orthogonal projection Π_{A_{0}}^{1,⊥} = id − Π_{A_{0}}^{1} from L^{2}(Λ^{1} ⊗ 𝔤_{E}) onto (Ker(d_{A_{0}}^{∗}|_{L^{2}}))^{⊥}, as

d_{A_{0}}^{∗}(id − d_{A_{0}}G_{A_{0}}^{0}d_{A_{0}}^{∗}) = 0.

For ζ ∈ (Ker(d_{A_{0}}|_{L_{1}^{2}}))^{⊥} and b = d_{A_{0}}ζ ∈ Im(d_{A_{0}}|_{L_{1}^{2}}), we have

∥G_{A_{0}}^{0}d_{A_{0}}^{∗}b∥_{L_{1,A_{0}}^{2}} = ∥G_{A_{0}}^{0}Δ_{A_{0}}^{0}ζ∥_{L_{1,A_{0}}^{2}} = ∥Π_{A_{0}}^{0}ζ∥_{L_{1,A_{0}}^{2}} = ∥ζ∥_{L_{1,A_{0}}^{2}}
≤ ∥d_{A_{0}}ζ∥_{L^{2}} + ∥ζ∥_{L^{2}} ≤ (1 + ν_{0}^{−1/2}) ∥d_{A_{0}}ζ∥_{L^{2}}
= (1 + ν_{0}^{−1/2}) ∥b∥_{L^{2}}

and so G_{A_{0}}^{0}d_{A_{0}}^{∗} has Hom(L^{2}, L_{1,A_{0}}^{2}) operator norm bound

∥G_{A_{0}}^{0}d_{A_{0}}^{∗}∥ ≤ 1 + ν_{0}^{−1/2}.

The Sobolev embedding L_{1}^{2} ⊂ L^{4} and Kato’s inequality imply that

∥d_{A}ζ − d_{A_{0}}ζ∥_{L^{2}} ≤ ∥[a_{0}, ζ]∥_{L^{2}} ≤ 2∥a_{0}∥_{L^{4}}∥ζ∥_{L^{4}} ≤ 2∥a_{0}∥_{L^{4}}∥ζ∥_{L_{1,A_{0}}^{2}} ,

and so d_{A} − d_{A_{0}} has Hom(L_{1,A_{0}}^{2}, L^{2}) operator norm bound

∥d_{A} − d_{A_{0}}∥ ≤ 2∥a_{0}∥_{L^{4}}.

In particular, we see that (DΨ)_{(id_{E},A_{0})}^{−1} = G_{A_{0}}^{0}d_{A_{0}}^{∗} ⊕ id = G_{A_{0}}^{0}d_{A_{0}}^{∗} ⊕ G_{A_{0}}^{0}Δ_{A_{0}}^{0} satisfies

∥(DΨ)_{(id_{E},A_{0})}^{−1}∥ ≤ 1+ν_{0}^{−1/2} and ∥(DΨ)_{(id_{E},A)} − (DΨ)_{(id_{E},A_{0})}∥ ≤ 2∥a_{0}∥_{L^{4}}.

Hence, Lemma 3.3 implies that if ∥a_{0}∥_{L^{4}} < \frac{1}{2}(1+ν_{0}^{−1/2})^{−1}, then the operator

(DΨ)_{(id_{E},A)} : (Ker (d_{A_{0}}|_{L_{1}^{2}}))^{⊥} × Ker (d_{A_{0}}^{∗}|_{L_{1}^{2}}) → L^{2}(Λ^{1} ⊗ 𝔤_{E})

is an isomorphism from L_{1}^{2} to L^{2} and restricts to a bounded linear map from L_{k+1}^{2} to L_{k}^{2}. Provided (DΨ)_{(id_{E},A)} : L_{k+1}^{2} → L_{k}^{2} is bijective, the open mapping theorem guarantees the existence of a bounded inverse (DΨ)_{(id_{E},A)}^{−1} : L_{k}^{2} → L_{k+1}^{2}. If (DΨ)_{(id_{E},A)}(ζ, a) = 0 for (ζ, a) ∈ L_{k+1}^{2}, then (ζ, a) is zero in L_{1}^{2} and thus zero in L_{k+1}^{2}, so (DΨ)_{(id_{E},A)} is injective. If b ∈ L_{k}^{2}(Λ^{1} ⊗ 𝔤_{E}), then b = (DΨ)_{(id_{E},A)}(ζ, a) = −d_{A}ζ + a for some (ζ, a) ∈ (Ker(d_{A_{0}}|_{L_{1}^{2}}))^{⊥} × Ker(d_{A_{0}}^{∗}|_{L_{1}^{2}}). As d_{A_{0}}^{∗}a = 0, we have

d_{A_{0}}^{∗}d_{A}ζ = −d_{A_{0}}^{∗}b ∈ L_{k−1}^{2}

and d_{A_{0}}^{∗}d_{A} : L_{k+1}^{2} → L_{k−1}^{2} is an elliptic operator with L_{k−1}^{2} coefficients. Thus, ζ ∈ L_{k+1}^{2}, so a = b + d_{A}ζ ∈ L_{k}^{2}, and (DΨ)_{(id_{E},A)} is surjective.

Combining the above observations, we see that the operator

(DΨ)_{(id_{E},A)} : (Ker (d_{A_{0}}|_{L_{k+1}^{2}}))^{⊥} ⊕ Ker (d_{A_{0}}^{∗}|_{L_{k}^{2}}) → L_{k}^{2}(Λ^{1} ⊗ 𝔤_{E}),

is an isomorphism for all A = A_{0}+a_{0} with ∥a_{0}∥_{L^{4}} < ε_{0} = \frac{1}{2}(1+ν_{0}^{−1/2})^{−1}. So, by the Banach space implicit function theorem, there are positive constants ε = ε(A, k) and δ = δ(A, k) and an open neighborhood U_{A} ⊂ A_{E}^{k} such that the map

Ψ : B_{id_{E}}^{⊥}(δ) × B_{A}(ε) → U_{A}, (u, A_{1}) ↦ u(A_{1}),

with B_{A}(ε) ⊂ B_{A_{0}}^{4}(ε_{0}), gives a diffeomorphism from an open neighborhood of (id_{E}, A) onto an open neighborhood of A. In particular, we obtain a map U_{A} → Stab_{A_{0}}^{⊥}, given by A_{1} ↦ u = u_{A_{1}}, such that

Ψ^{−1}(A_{1}) = (u, u^{−1}(A_{1})) ∈ B_{id_{E}}^{⊥}(δ) × B_{A}(ε) ⊂ Stab_{A_{0}}^{⊥} ×B_{A_{0}}^{4}(ε_{0}).

Hence, for any A_{1} ∈ U_{A} there is a unique u ∈ B_{id_{E}}^{⊥}(δ) such that u^{−1}(A_{1}) − A_{0} ∈ K_{A_{0}}:

d_{A_{0}}^{∗}(u^{−1}(A_{1}) − A_{0}) = 0. (3.5)

The neighborhood B_{A_{0}}^{4}(ε_{0}) is Stab_{A_{0}}-invariant: If A ∈ B_{A_{0}}(ε) and γ ∈ Stab_{A_{0}}, then

∥γ(A) − A_{0}∥_{L^{4}} = ∥A − γ^{−1}(A_{0})∥_{L^{4}} = ∥A − A_{0}∥_{L^{4}} < ε,

and

d_{A_{0}}^{∗} (γ(A) − A_{0}) = γ (d_{γ^{−1}(A_{0})}^{∗}(A − γ^{−1}(A_{0})))
= γ (d_{A_{0}}^{∗}(A − A_{0})) = 0,

so γ(A) ∈ B_{A_{0}}(ε).

The group G_{E}^{k+1} acts on G_{E}^{k+1} × S_{A_{0}} by (u, A) ↦ (vu, A), and so gives a diffeomorphism

B_{id_{E}}(δ) × B_{A_{0}}^{4}(ε_{0}) → B_{v}(δ) × B_{A_{0}}^{4}(ε_{0}), (u, A) → (vu, A),

and as this action commutes with the given action of Stab_{A_{0}}, it descends to a diffeomorphism

B_{id_{E}}(δ) ×_{Stab_{A_{0}}} B_{A_{0}}^{4}(ε_{0}) → B_{v}(δ) ×_{Stab_{A_{0}}} B_{A_{0}}^{4}(ε_{0}), [u, A] → [vu, A],

for each v ∈ G_{E}^{k+1}. Consequently, the G_{E}^{k+1}-equivariant map

G_{E}^{k+1} ×_{Stab_{A_{0}}} B_{A_{0}}^{4}(ε_{0}) → A_{E}^{k}

is a local diffeomorphism onto its image, as desired. □

Plainly, [γ(A)] = [A] for each γ ∈ Stab_{A_{0}} and A ∈ B_{A_{0}}^{4}(ε_{0}) and hence, the projection π : B_{A_{0}}^{4}(ε_{0}) → A_{E}^{k} factors through B_{A_{0}}^{4}(ε)/ Stab_{A_{0}}.

Lemma 3.7. There is a positive constant z with the following significance. Let ν_{0}[A_{0}] be the least positive eigenvalue of the Laplacian Δ_{A_{0}}^{0}. Then for any constant ε_{0} satisfying 0 < ε_{0} < z(1 + ν_{0}[A_{0}]^{−1/2})^{−1}, the projection map π : B_{A_{0}}^{4}(ε_{0})/ Stab_{A_{0}} → B_{E}^{k} is injective.

Proof. Suppose A_{i} ∈ B_{A_{0}}^{4}(ε_{0}) for i = 1, 2 and that [A_{1}] = [A_{2}] ∈ B_{E}^{k}, so u(A_{1}) = A_{2} for some u ∈ G_{E}^{k+1}. Since u(A_{0}) = A_{0} − (d_{A_{0}}u)u^{−1}, we see that u ∈ Stab_{A_{0}} if and only d_{A_{0}}u = 0. Here, we view u ∈ L_{k+1}^{2}(𝔤𝔩(E)) via the isometric embedding G_{E}^{k+1} ⊂ L_{k+1}^{2}(𝔤𝔩(E)) and write

u = u_{0} − γ,

where u_{0} ∈ (Ker d_{A_{0}})^{⊥} and γ ∈ Ker d_{A_{0}}. We claim that u_{0} = 0, so u = γ ∈ Stab_{A_{0}}.

Since u(A_{1}) := A_{1} − (d_{A_{1}}u)u^{−1} = A_{2}, we have A_{2}u = A_{1}u − d_{A_{1}}u = A_{1}u − d_{A_{0}}u − [A_{1} − A_{0}, u], and therefore

d_{A_{0}}u_{0} = d_{A_{0}}u = u(A_{1} − A_{0}) − (A_{2} − A_{0})u.

Since d_{A_{0}}^{∗}(A_{i} − A_{0}) = 0 for i = 1, 2, we obtain

d_{A_{0}}^{∗}d_{A_{0}}u_{0} = − ∗ (d_{A_{0}}u ∧ ∗(A_{1} − A_{0})) + ud_{A_{0}}^{∗}(A_{1} − A_{0})
− (d_{A_{0}}^{∗}(A_{2} − A_{0}))u + ∗(∗(A_{2} − A_{0}) ∧ d_{A_{0}}u)
= − ∗ (d_{A_{0}}u_{0} ∧ ∗(A_{1} − A_{0})) + ∗(∗(A_{2} − A_{0}) ∧ d_{A_{0}}u_{0}).

Integrating by parts gives

∥d_{A_{0}}u_{0}∥_{L^{2}}^{2} = (d_{A_{0}}^{∗}d_{A_{0}}u_{0}, u_{0})_{2} ≤ ∥d_{A_{0}}^{∗}d_{A_{0}}u_{0}∥_{L^{4/3}}∥u_{0}∥_{L^{4}}.

Kato’s inequality and the embedding L_{1}^{2} ⊂ L^{4} gives ∥u_{0}∥_{L^{4}} ≤ c(∥d_{A_{0}}u_{0}∥_{L^{2}} + ∥u_{0}∥_{L^{2}}), so the eigenvalue estimate ∥u_{0}∥_{L^{2}} ≤ ν_{0}^{−1/2}∥d_{A_{0}}u_{0}∥_{L^{2}} gives ∥u_{0}∥_{L^{4}} ≤ c(1 + ν_{0}^{−1/2})∥d_{A_{0}}u_{0}∥_{L^{2}} and thus

∥d_{A_{0}}u_{0}∥_{L^{2}}^{2} ≤ (1 + ν_{0}^{−1/2}) ∥d_{A_{0}}^{∗}d_{A_{0}}u_{0}∥_{L^{4/3}}∥d_{A_{0}}u_{0}∥_{L^{2}}.

Therefore, if d_{A_{0}}u_{0} ≢ 0, the preceding expression for d_{A_{0}}^{∗}d_{A_{0}}u_{0} yields

∥d_{A_{0}}u_{0}∥_{L^{2}} ≤ c (1 + ν_{0}^{−1/2}) ∥d_{A_{0}}^{∗}d_{A_{0}}u_{0}∥_{L^{4/3}}

≤ c (1 + ν_{0}^{−1/2}) ∥d_{A_{0}}u_{0}∥_{L^{2}}(∥A_{1} − A_{0}∥_{L^{4}} + ∥A_{2} − A_{0}∥_{L^{4}}),

and so we have 1 ≤ c (1 + ν_{0}^{−1/2}) (∥A_{1} − A_{0}∥_{L^{4}} + ∥A_{2} − A_{0}∥_{L^{4}}) ≤ c (1 + ν_{0}^{−1/2}) ε_{0}

which gives a contradiction for ε_{0} < c^{−1} (1 + ν_{0}^{−1/2})^{−1}. □

We now return to consider the local diffeomorphism Ψ of Lemma 3.6. Suppose Ψ[u_{1}, A_{1}] = Ψ[u_{2}, A_{2}] ∈ A_{E}^{k}, where [u_{1}, A_{1}], [u_{2}, A_{2}] ∈ G_{E}^{k+1}×_{Stab_{A_{0}}} B_{A_{0}}^{4}(ε_{0}), and so u_{1}(A_{1}) = u_{2}(A_{2}) ∈ A_{E}^{k} and hence [A_{1}] = [A_{2}] ∈ B_{E}^{k}. Provided ε_{0} also satisfies the constraints of Lemma 3.7, we have u_{2}^{−1}u_{1} = γ ∈ Stab_{A_{0}} and γ(A_{1}) = A_{2}. Hence [u_{2}, A_{2}] = [u_{1}γ^{−1}, γ(A_{1})] = [u_{1}, A_{1}], so Ψ is injective and therefore a diffeomorphism onto A_{E}^{k}.

The map π : B_{A_{0}}^{4}(ε_{0})/ Stab_{A_{0}} → B_{E}^{k} can be factored as the composition of the inclusion A ↦ (id_{E}, A) of B_{A_{0}}^{4}(ε_{0}) into G_{E}^{k+1} × B_{A_{0}}^{4}(ε_{0}), the projection onto the Stab_{A_{0}}-quotient G_{E}^{k+1} ×_{Stab_{A_{0}}} B_{A_{0}}^{4}(ε_{0}), the diffeomorphism Ψ of G_{E}^{k+1} ×_{Stab_{A_{0}}} B_{A_{0}}^{4}(ε_{0}) with A_{E}^{k} and the projection from A_{E}^{k} onto the G_{E}^{k+1}- quotient B_{E}^{k} = A_{E}^{k}/G_{E}^{k+1}. Hence, π is a homeomorphism onto an open neighborhood of [A_{0}] in B_{E}^{k} and a diffeomorphism on the open subset where Stab_{A_{0}} / Center(G) acts freely.

Claim 3.8. The quotient space B_{E}^{k} is Hausdorff.

Proof. Let Γ be the subspace {{A, u(A)} : A ∈ A_{E}^{k} and u ∈ G_{E}^{k+1}} of A_{E}^{k} × A_{E}^{k}. If {(A_{α}), u_{α}(A_{α})} is a sequence in Γ which converges in L_{k}^{2} to a point {A_{∞}, B_{∞}}, then Lemma 3.1 implies that there is a subsequence {α′} ⊂ {α} such that {u_{α}} converges in L_{k+1}^{2} to u_{∞} ∈ G_{E}^{k+1} and u_{∞}(A_{∞}) = B_{∞}. Thus, Γ is closed and A_{E}^{k}/G_{E}^{k+1} is Hausdorff. □

Claim 3.8 gives Assertion (1) of the proposition and Assertions (2), (3), and (4) now follow from the preceding arguments and Lemma 3.7. This completes the proof of the proposition. □

4. Critical-exponent Sobolev norms.

We now describe the basic properties of the critical-exponent norms and corresponding Banach spaces introduced by Taubes in [21], [23], [24], [25]. In particular, we give the basic embedding, multiplication, and composition lemmas we need to complete the proof of our slice theorem. We shall make frequent use of the pointwise Kato inequality, |d|v|| ≤ |∇_{A}v| for v ∈ Ω^{0}(E), so that the norms of the embedding and multiplication maps depend at most on the Riemannian manifold (X, g). Moreover, for simplicity, we confine our attention to the case of closed four-manifolds: There are obvious analogues of the Sobolev lemmas described here for any n-manifold, with n > 2. Similarly, extensions are possible to the case of complete manifolds bounded geometry (bounded curvature and injectivity radius uniformly bounded from below)—see [1], [3] for further details for Sobolev embedding results in those situations and for the construction of Green kernels. We refer the reader to the monograph of R. Adams [1] for a comprehensive treatment of Sobolev spaces and to that of E. Stein [19] for a treatment based on potential functions.

Throughout this section, A, B denote C^{∞} orthogonal connections on Riemannian vector bundles E, F over X with C^{∞} sections u, v, respectively. We first have the following analogues of the L^{2} and L^{4} norms,

∥u∥_{L^{♯}(X)} = sup_{x∈X} ∥ dist^{−2}(x, ·)|u|∥_{L^{1}(X)},
∥u∥_{L^{2♯}(X)} = sup_{x∈X} ∥ dist^{−1}(x, ·)|u|∥_{L^{2}(X)}, (4.1)

where dist(x, y) denotes the geodesic distance between points x and y in X defined by the metric g; these norms have the same behavior as the L^{2} and L^{4} norms with respect to constant rescalings of the metric g—the L^{♯} norm on two-forms and the L^{2♯} norm on one-forms are scale invariant. Indeed, one sees this by noting that if g ↦ g̃ = λ^{−2}g, then distg̃(x, y) = λ^{−1} dist_{g}(x, y) and dVg̃ = λ^{−4}dV_{g}, while for any a ∈ Ω^{1}(E) and v ∈ Ω^{2}(E), we have |a|g̃ = λ|a|_{g}, and |v|g̃ = λ^{2}|v|_{g}.

Next, we define analogues of the L_{1}^{2} and L_{2}^{2} norms

∥u∥_{L_{1,A}^{2}(X)} = ∥∇_{A}u∥_{L^{2}(X)} + ∥u∥_{L^{2}(X)},
∥u∥_{L_{2,A}^{2}(X)} = ∥∇_{A}^{2}u∥_{L^{2}(X)} + ∥∇_{A}u∥_{L^{2}(X)} + ∥u∥_{L^{2}(X)},

and set

\begin{aligned}∥u∥_{L_{1,A}^{♯}(X)} & = ∥∇_{A}u∥_{L^{♯}(X)} + ∥u∥_{L^{2♯}(X)} + ∥u∥_{L^{♯}(X)}, \\ ∥u∥_{L_{2,A}^{♯}(X)} & = ∥∇_{A}^{∗}∇_{A}u∥_{L^{♯}(X)} + ∥u∥_{L^{♯}(X)}, \end{aligned} (4.2)

where ∇_{A}^{∗} = − ∗ ∇_{A}∗ : Ω^{1}(E) → Ω^{0}(E) is the L^{2}-adjoint of the map ∇_{A} : Ω^{0}(E) → Ω^{1}(E).

Finally, we define analogues of the C^{0} ∩ L_{2}^{2} norm

∥u∥_{C^{0}∩L_{2,A}^{2}(X)} = ∥u∥_{C^{0}(X)} + ∥u∥_{L_{2,A}^{2}(X)},

and set

\begin{aligned}∥u∥_{L^{♯,2}(X)} & = ∥u∥_{L^{♯}∩L^{2}(X)} = ∥u∥_{L^{♯}(X)} + ∥u∥_{L^{2}(X)}, \\ ∥u∥_{L^{2♯,4}(X)} & = ∥u∥_{L^{2♯}∩L^{4}(X)} = ∥u∥_{L^{2♯}(X)} + ∥u∥_{L^{4}(X)}, \\ ∥u∥_{L_{1,A}^{♯,2}(X)} & = ∥u∥_{L_{1,A}^{♯}∩L_{1,A}^{2}(X)} = ∥u∥_{L_{1,A}^{♯}(X)} + ∥u∥_{L_{1,A}^{2}(X)}, \\ ∥u∥_{L_{2,A}^{♯,2}(X)} & = ∥u∥_{L_{2,A}^{♯}∩L_{2,A}^{2}(X)} = ∥u∥_{L_{2,A}^{♯}(X)} + ∥u∥_{L_{2,A}^{2}(X)}. \end{aligned} (4.3)

It might have appeared, at first glance, a little more natural to continue the obvious pattern and instead define ∥u∥_{L_{2,A}^{♯}(X)} using ∥∇_{A}^{2}u∥_{L^{♯}(X)}: As we shall see below, though, the given definition is most useful in practice. For related reasons, if u ∈ Ω^{1}(E) = Ω^{0}(Λ^{1} ⊗ E), it is convenient to define the norm ∥u∥_{L_{1,A}^{♯}(X)} by

∥u∥_{L_{1,A}^{♯}(X)} = ∥∇_{A}^{∗}u∥_{L^{♯}(X)} + ∥u∥_{L^{2♯}(X)} + ∥u∥_{L^{♯}(X)}. (4.4)

Let L^{♯}(X) be the Banach space completion of C^{∞}(X) with respect to the norm ∥ · ∥_{L^{♯}} and similarly define the remaining Banach spaces above.

We have the following extensions of the standard Sobolev embedding theorem [10], [15]: Their proofs are given in the next section. See also [4], [17], [21], [23, §6], [24, Eq. (3.4) & §5], and [25, Lemma 4.7].

Lemma 4.1. The following are continuous embeddings:

(1) L_{k}^{p}(E) ⊂ L_{k}^{♯}(E), for k = 0, 1, 2 and all p > 2;

(2) L^{q}(E) ⊂ L^{2♯}(E), for all q > 4;

(3) L_{1}^{2}(E) ⊂ L^{2♯}(E).

In the reverse direction we have:

Lemma 4.2. The following are continuous embeddings:

(1) L^{♯}(E) ⊂ L^{1}(E) and L^{2♯}(E) ⊂ L^{2}(E);

(2) L_{2}^{♯}(E) ⊂ C^{0} ∩ L_{1}^{2}(E).

We next consider the extension of the standard Sobolev multiplication lemma [10], [15]. While there is no continuous multiplication map L_{2}^{2} × L_{2}^{2} → L_{2}^{2}, it is worth observing that there is a continuous bilinear map C^{0} ∩ L_{2}^{2}(E) × C^{0} ∩ L_{2}^{2}(F) → C^{0} ∩ L_{2}^{2}(E ⊗ F) given by (u, v) ↦ u ⊗ v. Note that for u ∈ Ω^{0}(E) and v ∈ Ω^{0}(F) we have

(4.5)  ∇_{A⊗B}^{2}(u ⊗ v) = (∇_{A}^{2}u) ⊗ v + 2∇_{A}u ⊗ ∇_{B}v + u ⊗ ∇_{B}^{2}v,
∇_{A⊗B}^{∗}∇_{A⊗B}(u ⊗ v) = (∇_{A}^{∗}∇_{A}u) ⊗ v + ∗((∗∇_{A}u) ∧ ∇_{B}v)
− ∗(∇_{A}u ∧ ∗∇_{B}v) + u ⊗ ∇_{B}^{∗}∇_{B}v.

Similarly, for u ∈ Ω^{0}(Λ^{1} ⊗ E) and v ∈ Ω^{0}(F), we have

∇_{A⊗B}^{∗}(u ⊗ v) = (∇_{A}^{∗}u) ⊗ v + ∗(∗u ∧ ∇_{B}v). (4.6)

In particular, we see that if u, v ∈ Ω^{0}(𝔤𝔩(E)), then

(4.7)  ∇_{A}^{∗}∇_{A}(uv) = (∇_{A}^{∗}∇_{A}u)v + ∗((∗∇_{A}u) ∧ ∇_{A}v)
− ∗(∇_{A}u ∧ (∗∇_{A}v)) + u(∇_{A}^{∗}∇_{A}v),

an identity we will need in the next section.

Lemma 4.3. Let Ω^{0}(E) × Ω^{0}(F) → Ω^{0}(E ⊗ F) be given by (u, v) ↦ u ⊗ v. Then the following hold.

(1) The map C^{0}(E) ⊗ L^{♯}(F) → L^{♯}(E ⊗ F) is continuous;

(2) The map L^{2♯}(E) ⊗ L^{2♯}(F) → L^{♯}(E ⊗ F) is continuous;

(3) The spaces L_{1}^{♯}(F), L_{1}^{2}(F), and L_{2}^{♯}(F) are L_{2}^{♯}(E)-modules;

(4) The spaces L_{1}^{2}(F), L_{1}^{♯,2}(F), and L_{2}^{♯,2}(F) are L_{2}^{♯,2}(E)-modules. The conclusions continue to hold for Ω^{1}(E) in place of Ω^{0}(E) and the norms on L_{1}^{♯}(Λ^{1} ⊗ E) and L_{1}^{♯,2}(Λ^{1} ⊗ E) defined via (4.4).

Proof. Let u ∈ C^{∞}(E) and v ∈ C^{∞}(F) and denote the covariant derivatives on E, F, and E ⊗ F by ∇. Using ∇(u ⊗ v) = (∇u)v + u ⊗ ∇v and the embedding L_{2}^{♯}(E) ⊂ C^{0}(E), we see that

∥u ⊗ v∥_{L^{♯}} ≤ ∥u∥_{C^{0}}∥v∥_{L^{♯}} and ∥u ⊗ v∥_{L^{♯}} ≤ ∥u∥_{L^{2♯}}∥v∥_{L^{2♯}},
∥∇(u ⊗ v)∥_{L^{♯}} ≤ ∥∇u∥_{L^{2♯}}∥v∥_{L^{2♯}} + ∥u∥_{C^{0}}∥∇v∥_{L^{♯}} ≤ c∥u∥_{L_{2}^{♯}}∥v∥_{L_{1}^{♯}},
∥∇(u ⊗ v)∥_{L^{2}} ≤ ∥∇u∥_{L^{4}}∥v∥_{L^{4}} + ∥u∥_{C^{0}}∥∇v∥_{L^{2}} ≤ ∥u∥_{C^{0}∩L_{2}^{2}}∥v∥_{L_{1}^{2}},

and hence the multiplication maps C^{0} × L^{♯} → L^{♯}, L^{2♯} × L^{2♯} → L^{♯}, and L_{2}^{♯} × L_{1}^{♯} → L_{1}^{♯} are continuous. Moreover,

∥∇(u ⊗ v)∥_{L^{2}} ≤ ∥∇u∥_{L^{4}}∥v∥_{L^{4}} + ∥u∥_{C^{0}}∥∇v∥_{L^{2}} ≤ c∥u∥_{C^{0}∩L_{2}^{2}}∥v∥_{L_{1}^{2}},

and so, using the embedding L_{2}^{♯} ⊂ C^{0}, the multiplication L_{2}^{♯,2} × L_{1}^{♯,2} → L_{1}^{♯,2} is continuous. Thus, L_{2}^{♯} is an L_{1}^{♯}-module and L_{2}^{♯,2} is an L_{1}^{♯,2}-module.

Finally, to see that L_{2}^{♯} and L_{2}^{♯,2} are algebras, we use the identities (4.5), noting that

∥∇^{∗}∇(u ⊗ v)∥_{L^{♯}} ≤ ∥∇^{∗}∇u∥_{L^{♯}}∥v∥_{C^{0}} + 2∥∇u∥_{L^{2♯}}∥∇v∥_{L^{2♯}} + ∥u∥_{C^{0}}∥∇^{∗}∇v∥_{L^{♯}}
≤ c∥u∥_{L_{2}^{♯}}∥v∥_{L_{2}^{♯}},

so the multiplication L_{2}^{♯} × L_{2}^{♯} → L_{2}^{♯} is continuous, while

∥∇^{2}(u ⊗ v)∥_{L^{2}} ≤ ∥∇^{2}u∥_{L^{2}}∥v∥_{C^{0}} + 2∥∇u∥_{L^{4}}∥∇v∥_{L^{4}} + ∥u∥_{C^{0}}∥∇^{2}v∥_{L^{2}}
≤ c∥u∥_{C^{0}∩L_{2}^{2}}∥v∥_{C^{0}∩L_{2}^{2}}.

The embedding L_{2}^{♯} ⊂ C^{0} now implies that the multiplication L_{2}^{♯,2} × L_{2}^{♯,2} → L_{2}^{♯,2} is continuous. □

5. Critical-exponent Sobolev embeddings and estimates for Green’s operators.

We continue the notation and assumptions of Section 4. Our goal in this section is to prove the Sobolev embedding Lemmas 4.1 and 4.2, and to derive estimates for the Green’s operator G_{A} of the Laplacian ∇_{A}^{∗}∇_{A} on Ω^{0}(E). The key estimates described in this section are due to Taubes and they arise, in a variety of contexts, in the proofs of [17, Lemma 5.4], [21, Equation (2.14) & Lemmas 3.5, 3.6, & A.3], [23, Equation (3.4b) & Lemma 6.2], [24, Lemma 5.6], and [25, §4(c), (d), (e)]. However, we find it convenient to collect them here—together with some useful extensions and generalizations—both for the purposes of the present article and applications in [8], [9]. 5.1. Estimates for the covariant Laplacian ∇_{A}^{∗}∇_{A}. Let G(x, y) be the kernel function for the Green’s operator (d^{∗}d+1)^{−1} of the Laplacian d^{∗}d+1 on C^{∞}(X). The kernel G(x, y) of (d^{∗}d + 1)^{−1} behaves like dist^{−2}(x, y) as dist(x, y) → 0 (see [25, Lemma 4.7] and [24, §5]):

Lemma 5.1. The kernel G(x, y) is a positive C^{∞} function away from the diagonal in X × X and as dist(x, y) → 0,

G(x, y) = \frac{1}{4π^{2} dist^{2}(x, y)} + o(dist^{−2}(x, y)).

Proof. These and other properties of G are obtained by explicitly constructing G from an initial choice of parametrix H for d^{∗}d + 1 using the method of [3, §4.2.2–3], where the kernel for the Green’s operator for d^{∗}d is constructed. Recall from [19, p. 132] that the kernel G_{0}(x, y) for (d^{∗}d + 1)^{−1} on ℝ^{4} with its standard metric satisfies

G_{0}(x, y) = \frac{1}{4π^{2}|x − y|^{2}} + o(|x − y|^{−2}), |x − y| → 0.

The kernel G is now constructed using G_{0} by following the method of [3, §4.2.2–3]. □

Lemma 5.1 implies that there is a constant c depending at most on g such that for all x ≠ y in X,

(5.1) c^{−1} dist^{−2}(x, y) ≤ G(x, y) ≤ c dist^{−2}(x, y).

Consequently, for all u ∈ Ω^{0}(E), we have

c^{−1}∥u∥_{L^{♯}(X)} ≤ ∥G|u|∥_{C^{0}(X)} ≤ c∥u∥_{L^{♯}(X)}. (5.2)

Lemma 4.2 will follow from the next estimate; a similar inequality is stated as Equation (3.4) in [24]; see [17, Lemma 5.4(a)] for a related result on ℝ^{3}.

Lemma 5.2. For all f ∈ L_{1}^{2}(ℝ^{4}), where ℝ^{4} has its standard metric,

sup_{x∈ℝ^{4}} ∥ dist^{−1}(x, ·)f∥_{L^{2}(ℝ^{4})} ≤ \frac{1}{2}∥∇f∥_{L^{2}(ℝ^{4})}.

Suppose X be a closed, oriented, Riemannian four-manifold. Then there is a positive constant c such that for all f ∈ L_{1}^{2}(X),

sup_{x∈X} ∥ dist^{−1}(x, ·)f∥_{L^{2}(X)} ≤ c∥f∥_{L_{1}^{2}(X)}.

Proof. Let f ∈ C_{0}^{∞}(ℝ^{4}) and let x = (r, θ) denote polar coordinates centered at a point x_{0} ∈ ℝ^{4}, so r = |x − x_{0}|. Then

∫_{ℝ^{4}} r^{−2}|f|^{2} dx = ∫_{S^{3}} ∫_{ℝ} rf^{2} drdθ

= \frac{1}{2} ∫_{S^{3}} ∫_{ℝ} \frac{dr^{2}}{dr} f^{2} drdθ = −\frac{1}{2} ∫_{S^{3}} ∫_{ℝ} r^{2}f \frac{∂f}{∂r} drdθ,

via integration by parts. Therefore,

∫_{ℝ^{4}} r^{−2}|f|^{2} dx = −\frac{1}{2} ∫_{ℝ^{4}} r^{−1}f \frac{∂f}{∂r} dx

≤ \frac{1}{2} (∫_{ℝ^{4}} r^{−2}|f|^{2} dx)^{1/2} (∫_{ℝ^{4}} |∇f|^{2} dx)^{1/2} .

Hence, for all f ∈ C_{0}^{∞}(ℝ^{4}) we have

∥ dist^{−1}(x_{0}, ·)f∥_{L^{2}(ℝ^{4})} ≤ \frac{1}{2}∥∇f∥_{L^{2}(ℝ^{4})},

and taking the supremum over x_{0} ∈ ℝ^{4} yields the first assertion.

For a closed Riemannian manifold X, choosing a smooth partition of unity for X and applying first assertion (when X is ℝ^{4}) to each patch then yields the second assertion. □

Proof of Lemma 4.1. Define 1 ≤ p′ < 2 by setting 1 = 1/p + 1/p′. Then Hölder’s inequality implies that

∥ dist^{−2}(x, ·)|u|∥_{L^{1}} ≤ ∥ dist^{−2}(x, ·)∥_{L^{p′}} ∥u∥_{L^{p}} ≤ C∥u∥_{L^{p}},
∥ dist^{−1}(x, ·)|u|∥_{L^{2}} ≤ ∥ dist^{−1}(x, ·)∥_{L^{2p′}} ∥u∥_{L^{2p}} ≤ C∥u∥_{L^{2p}},

which gives Assertions (1) and (2). By Lemma 5.2 and Kato’s inequality, |d|u|| ≤ |∇_{A}u|, we see that

sup_{x∈X} ∥ dist^{−1}(x, ·)u∥_{L^{p}} = sup_{x∈X} ∥ dist^{−1}(x, ·)|u|∥_{L^{p}}
≤ C(∥d|u|∥_{L^{p}} + ∥u∥_{L^{p}})
≤ C(∥∇_{A}u∥_{L^{p}} + ∥u∥_{L^{p}}).

Taking p = 2 gives Assertion (3). □

Lemma 4.2 will follow from the estimates below; the key estimates (1) and (2) in Lemma 5.3 below and the estimates (1), (2), and (3) in Lemma 5.4 are essentially those of Lemma 6.2 in [23], except that the dependence of the constant on ∥F_{A}∥_{L^{2}} is made explicit, but the argument is the same as that of [23].

Lemma 5.3. Let X be a closed, oriented four-manifold with metric g. Then there is a constant c with the following significance. Let E be a Riemannian vector bundle over X and let A be an orthogonal L_{2}^{2} connection on E with curvature F_{A}. Then L_{2}^{♯}(E) ⊂ C^{0} ∩ L_{1}^{2}(E) and the following estimates hold:

(1)  ∥∇_{A}u∥_{L^{2♯}(X)} + ∥u∥_{C^{0}(X)} ≤ c∥∇_{A}^{∗}∇_{A}u∥_{L^{♯}(X)} + ∥u∥_{L^{♯}(X)},
(2)  ∥∇_{A}u∥_{L^{2♯}(X)} + ∥u∥_{C^{0}(X)} ≤ c∥∇_{A}^{∗}∇_{A}u∥_{L^{♯}(X)} + ∥u∥_{L^{2}(X)},
(3)  ∥u∥_{L^{1}(X)} ≤ c∥u∥_{L^{♯}(X)},
(4)  ∥u∥_{L^{2}(X)} ≤ c∥u∥_{L^{2♯}(X)},
(5)  ∥∇_{A}u∥_{L^{2}(X)} ≤ c∥∇_{A}u∥_{L^{2♯}(X)}.

Proof. For any u ∈ C^{∞}(E) there is the following pointwise identity [10, p. 93],

|∇_{A}u|^{2} + \frac{1}{2}d^{∗}d|u|^{2} = ⟨∇_{A}^{∗}∇_{A}u, u⟩,

and thus:

|∇_{A}u|^{2} + \frac{1}{2}(1 + d^{∗}d)|u|^{2} = ⟨∇_{A}^{∗}∇_{A}u, u⟩ + \frac{1}{2}|u|^{2}.

Using this identity and the fact that ∫_{X} G(x, ·)(d^{∗}d + 1)|u|^{2} dV = |u|^{2}(x), we obtain

∫_{X} G(x, ·)|∇_{A}u|^{2} dV + \frac{1}{2}|u|^{2}(x)

≤ ∫_{X} G(x, ·)|⟨∇_{A}^{∗}∇_{A}u, u⟩| dV + \frac{1}{2} ∫_{X} G(x, ·)|u|^{2} dV.

Therefore, from (5.1), we have

∥|∇_{A}u|^{2}∥_{L^{♯}} + ∥|u|^{2}∥_{C^{0}} ≤ c∥⟨∇_{A}^{∗}∇_{A}u, u⟩∥_{L^{♯}} + c∥|u|^{2}∥_{L^{♯}}
≤ c∥∇_{A}^{∗}∇_{A}u∥_{L^{♯}}∥u∥_{C^{0}} + c∥u∥_{L^{♯}}∥u∥_{C^{0}}.

Consequently, using rearrangement with the last term, we see that

∥∇_{A}u∥_{L^{2♯}} + ∥u∥_{C^{0}} ≤ c∥∇_{A}^{∗}∇_{A}u∥_{L^{♯}} + c∥u∥_{L^{♯}},

giving (1). Combining this estimate with the embedding and interpolation inequalities, ∥u∥_{L^{♯}} ≤ c∥u∥_{L^{4}} ≤ c∥u∥_{L^{2}}^{1/2}∥u∥_{C^{0}}^{1/2}, and again using rearrangement with the last term yields the bound in (2). Since X is closed, for all x ≠ y we have dist(x, y) ≤ M < ∞, so

∫_{X} dist^{−2}(x, ·)|u| dV ≥ M^{−2} ∫_{X} |u| dV,

and this gives the estimates in (3), (4), and (5). □

Proof of Lemma 4.2. From Lemma 5.3 we have the estimate

∥u∥_{C^{0}} ≤ c∥u∥_{L_{2,A_{0}}^{♯}} ,

for any u ∈ C^{∞}(E). Let {u_{m}} be a sequence in C^{∞}(E) converging to u ∈ L_{2}^{♯}(E). The sequence {u_{m}} is Cauchy in L_{2}^{♯}(E) and applying the preceding estimate to the differences u_{m_{2}} −u_{m_{1}}, we see that it is Cauchy in the Banach space C^{0}(E) and so the limit u lies in C^{0}(E). The same argument, with estimates (1) and (5) of Lemma 5.3, shows that u ∈ L_{1}^{2}(E) and this yields Assertion (2) of the lemma. Assertion (1) follows in the same manner. □

Lemma 5.4. Continue the hypotheses of Lemma 5.3. Then for any u ∈ (C^{0} ∩ L_{2}^{2})(E), we have

(1) ∥∇_{A}^{2}u∥_{L^{2}(X)} ≤ ∥∇_{A}^{∗}∇_{A}u∥_{L^{2}(X)} + c∥F_{A}∥_{L^{2}(X)}^{1/2}∥∇_{A}u∥_{L^{4}(X)}
+ ∥F_{A}∥_{L^{2}(X)}∥u∥_{C^{0}(X)},
(2) ∥∇_{A}u∥_{L^{4}(X)} ≤ ∥u∥_{C^{0}(X)}^{1/2} (∥∇_{A}^{∗}∇_{A}u∥_{L^{2}(X)} + 2∥∇_{A}^{2}u∥_{L^{2}(X)})^{1/2} ,
(3) ∥∇_{A}^{2}u∥_{L^{2}(X)} ≤ 2∥∇_{A}^{∗}∇_{A}u∥_{L^{2}(X)} + c∥F_{A}∥_{L^{2}(X)}∥u∥_{C^{0}(X)}.

Proof. The Bochner-Weitzenböck formula for the covariant Laplacian [11, Appendix, Theorem II.1] asserts that

d_{A}^{∗}d_{A} + d_{A}d_{A}^{∗} = ∇_{A}^{∗}∇_{A} + {F_{A}, ·}, (5.3)

where we use {·, ·} to denote a certain bilinear map whose precise form is unimportant here. Integrating by parts and noting that d_{A} = ∇_{A} and d_{A}^{∗}d_{A} = ∇_{A}^{∗}∇_{A} on Ω^{0}(X, V) and F_{A} = d_{A} ∘ d_{A} gives

∥∇_{A}^{2}u∥_{L^{2}}^{2} = (∇_{A}^{∗}∇_{A}∇_{A}u, ∇_{A}u)_{L^{2}}
= ((d_{A}^{∗}d_{A} + d_{A}d_{A}^{∗})d_{A}u, d_{A}u)_{L^{2}} − ({F_{A}, d_{A}u}, ∇_{A}u)_{L^{2}}

= (d_{A}^{∗}F_{A}u, d_{A}u)_{L^{2}} + (d_{A}(d_{A}^{∗}d_{A})u, d_{A}u)_{L^{2}} − ({F_{A}, ∇_{A}u}, ∇_{A}u)_{L^{2}} = (F_{A}u, F_{A}u)_{L^{2}} + (∇_{A}^{∗}∇_{A}u, ∇_{A}^{∗}∇_{A}u)_{L^{2}} − ({F_{A}, ∇_{A}u}, ∇_{A}u)_{L^{2}}.

Therefore, applying Hölder’s inequality, we find that

∥∇_{A}^{2}u∥_{L^{2}}^{2} ≤ ∥∇_{A}^{∗}∇_{A}u∥_{L^{2}}^{2} + c∥F_{A}∥_{L^{2}}∥∇_{A}u∥_{L^{4}}^{2} + ∥F_{A}∥_{L^{2}}^{2}∥u∥_{C^{0}}^{2},

and taking square roots gives the desired bound in (1).

We now use integration by parts and Kato’s inequality |d|u|| ≤ |∇_{A}u| to obtain an L^{4} bound on d_{A}u:

∥d_{A}u∥_{L^{4}}^{4} = (d_{A}u, |d_{A}u|^{2}d_{A}u)_{L^{2}}
= (u, |d_{A}u|^{2}d_{A}^{∗}d_{A}u)_{L^{2}} + 2 (u, |d_{A}u|d_{A}u ∧ d|d_{A}u|)_{L^{2}} ,
= ∥u∥_{C^{0}}∥d_{A}u∥_{L^{4}}^{2}∥d_{A}^{∗}d_{A}u∥_{L^{2}} + 2∥u∥_{C^{0}}∥d_{A}u∥_{L^{4}}^{2}∥∇_{A}d_{A}u∥_{L^{2}},

and so, if d_{A}u ≢ 0,

∥d_{A}u∥_{L^{4}} ≤ ∥u∥_{C^{0}}^{1/2} (∥d_{A}^{∗}d_{A}u∥_{L^{2}} + 2∥∇_{A}^{2}u∥_{L^{2}})^{1/2} ,

which gives the desired estimate in (2).

By combining the L^{4} estimate for ∇_{A}u with the L^{2} estimate for ∇_{A}^{2}u, we obtain

∥∇_{A}^{2}u∥_{L^{2}} ≤ ∥∇_{A}^{∗}∇_{A}u∥_{L^{2}} + ∥F_{A}∥_{L^{2}}∥u∥_{C^{0}}
+ c∥F_{A}∥_{L^{2}}^{1/2}∥u∥_{C^{0}}^{1/2} (∥∇_{A}^{∗}∇_{A}u∥_{L^{2}} + ∥∇_{A}^{2}u∥_{L^{2}})^{1/2} .

We now use rearrangement with the last term above to give

∥∇_{A}^{2}u∥_{L^{2}} ≤ 2∥∇_{A}^{∗}∇_{A}u∥_{L^{2}} + c∥F_{A}∥_{L^{2}}∥u∥_{C^{0}},

and this establishes the desired bound in (3). □

Lemma 5.5. Continue the hypotheses of Lemma 5.3. Then for any u ∈ L_{2}^{♯,2}(E), we have:

∥u∥_{L_{2,A}^{2}(X)} + ∥u∥_{C^{0}(X)} ≤ c(1 + ∥F_{A}∥_{L^{2}(X)}) (∥∇_{A}^{∗}∇_{A}u∥_{L^{♯,2}(X)} + ∥u∥_{L^{2}(X)}) .

Proof. From Assertion (3) of Lemma 5.4 we have the estimate

∥∇_{A}^{2}u∥_{L^{2}} ≤ 2∥∇_{A}^{∗}∇_{A}u∥_{L^{2}} + c∥F_{A}∥_{L^{2}}∥u∥_{C^{0}},

while integration by parts gives

∥∇_{A}u∥_{L^{2}} = (∇_{A}^{∗}∇_{A}u, u)_{L^{2}}^{1/2} ≤ \frac{1}{\sqrt{2}} (∥∇_{A}^{∗}∇_{A}u∥_{L^{2}} + ∥u∥_{L^{2}}) .

According to Lemma 5.3 we have

∥u∥_{C^{0}} ≤ c∥∇_{A}^{∗}∇_{A}u∥_{L^{♯}} + c∥u∥_{L^{2}},

and therefore the desired bound follows by combining these estimates. □

The above lemmas lead to the following estimates for the Green’s operator G_{A} : L^{♯,2}(E) → L_{2}^{♯,2}(E) of the Laplacian ∇_{A}^{∗}∇_{A} : L_{2}^{♯,2}(E) → L^{♯,2}(E). For u ∈ Ω^{0}(E) define

∥u∥_{L_{2,A}^{♯,2}(X)} = ∥∇_{A}^{∗}∇_{A}u∥_{L^{♯,2}(X)} + ∥u∥_{L^{♯,2}(X)}, (5.4)

and observe that this is equivalent to the L_{2,A}^{♯,2} norm defined in Section 4, although the comparison depends on the L^{2} norm of the curvature F_{A}.

Lemma 5.6. Continue the hypotheses of Lemma 5.3. Let ν_{0}[A] be the least positive eigenvalue of the Laplacian ∇_{A}^{∗}∇_{A}. Then for any u ∈ L^{♯,2} ∩ (Ker ∇_{A}^{∗}∇_{A})^{⊥}, we have:

(1)  ∥G_{A}u∥_{L_{2,A}^{♯}(X)} ≤ c(1 + ν_{0}[A]^{−1})∥u∥_{L^{♯,2}(X)},
(2)  ∥G_{A}u∥_{L_{2,A}^{♯,2}(X)} ≤ c(1 + ν_{0}[A]^{−1})∥u∥_{L^{♯,2}(X)},
(3)  ∥G_{A}u∥_{L_{2,A}^{♯,2}(X)} ≤ c(1 + ν_{0}[A]^{−1})(1 + ∥F_{A}∥_{L^{2}(X)})∥u∥_{L^{♯,2}(X)}.

Proof. The first and second assertions follow from Lemma 5.3, the fact that ∇_{A}^{∗}∇_{A}G_{A}u = u for u ∈ (Ker ∇_{A}^{∗}∇_{A})^{⊥}, and the eigenvalue estimate ∥u∥_{L^{2}} ≤ ν_{0}[A]^{−1}∥∇_{A}^{∗}∇_{A}u∥_{L^{2}}, while the third assertion follows from the first and Lemma 5.5. □

5.2. Elliptic estimates for d_{A}^{+} + d_{A}^{∗}. To illustrate their application and to point to possible extensions, we note that the estimates of Section 5.1 for the covariant Laplacian ∇_{A}^{∗}∇_{A} = d_{A}^{∗}d_{A} on Ω^{0}(E) naturally extend to give estimates for the covariant Laplacians ∇_{A}^{∗}∇_{A} on Ω^{ℓ}(E) = Ω^{0}(Λ^{ℓ} ⊗ E). Estimates for ∇_{A}^{∗}∇_{A} on Ω^{0}(Λ^{1}⊗E) and Ω^{0}(Λ^{+}⊗E) are of particular interest since these can in turn be profitably compared (via the Bochner-Weitzenböck formulas [10, Equations (6.25) & (6.26)], as in [24]) with the remaining Laplacians defined by the elliptic deformation complex for the anti-self-dual equation [5], [10], namely d_{A}d_{A}^{∗} + d_{A}^{+,∗}d_{A}^{+} on Ω^{0}(Λ^{1} ⊗ 𝔤_{E}) and d_{A}^{+}d_{A}^{+,∗} on Ω^{0}(Λ^{+} ⊗ 𝔤_{E}). Indeed, if B_{1} and B_{+} are the Levi-Civita connections on Λ^{1} and Λ^{+} induced by the Levi-Civita connection on T X for the metric g, then the curvature “F_{A}” in the estimates of the preceding subsection is simply replaced by [12, p. 165]

F_{B_{1}⊗A} = F_{B_{1}} ⊗ id_{𝔤_{E}} + id_{Λ^{1}} ⊗ F_{A},
F_{B_{+}⊗A} = F_{B_{+}} ⊗ id_{𝔤_{E}} + id_{Λ^{+}} ⊗ F_{A}, (5.5)

where F_{B_{1}} and F_{B_{+}} are expressed in terms of the Riemann curvature tensor Rm and where we abuse notation slightly and denote the connections on E and 𝔤_{E} both by A. (See [10, Appendix C] and [11, Appendix II].) In the interests of brevity we shall confine our attention to the case of L_{ℓ}^{p} estimates with p = 2, though the methods can be modified to obtain estimates for p ≠ 2 (some work is required—see [5, p. 426] for hints).

In order to compute the required elliptic estimates for d_{A}^{+} we will need the Bochner-Weitzenböck formulas,

(5.6)  d_{A}d_{A}^{∗} + 2d_{A}^{∗}d_{A}^{+} = ∇_{A}^{∗}∇_{A} + {Ric, ·} − 2{F_{A}^{−}, ·},
(5.7)  2d_{A}^{+}d_{A}^{∗} = ∇_{A}^{∗}∇_{A} − 2\{W^{+}, ·\} + \frac{R}{3} + \{F_{A}^{+}, ·\},

for the Laplacians on Ω^{1}(𝔤_{E}) and Ω^{+}(𝔤_{E}) [10, Equations (6.25) & (6.26)]; here, Ric, W^{+}, and R are the Ricci, self-dual Weyl, and scalar curvatures of the Riemannian metric g on X. In applications to the degeneration of anti-self-dual or “almost anti-self-dual” connections A as in [20], [22], [23], [24], we can usually arrange to have a uniform L^{∞} bound on F_{A}^{+}, but not a uniform L^{p} bound on F_{A}^{−} when p > 2. We derive estimates in the remainder of this subsection with such applications and assumptions in view. To illustrate the nature of the difficulty we first derive a naive L_{1,A}^{2} estimate for a ∈ L_{1}^{2}(𝔤_{E}) in terms of the L^{2} norm of (d_{A}^{∗} + d_{A}^{+})a:

Lemma 5.7. Let X be a closed, oriented four-manifold with metric g. Then there is a constant c with the following significance. Let E be a Riemannian vector bundle over X and let A be an orthogonal L_{4}^{2} connection on E with curvature F_{A}. Then for any a ∈ L_{1}^{2}(Λ^{1} ⊗ 𝔤_{E}),

\begin{aligned}∥a∥_{L_{1,A}^{2}(X)} & ≤ \sqrt{2}∥(d_{A}^{∗} + d_{A}^{+})a∥_{L^{2}(X)} \\ & + c (1 + ∥F_{A}^{−}∥_{C^{0}(X)})^{1/2} ∥a∥_{L^{2}(X)}. \end{aligned} (5.8)

If a is L^{2}-orthogonal to Ker d_{A}^{+}, so that a = d_{A}^{∗}v for some v ∈ L_{2}^{2}(Λ^{+} ⊗ 𝔤_{E}), then

\begin{aligned}∥d_{A}^{∗}v∥_{L_{1,A}^{2}(X)} & ≤ \sqrt{2}∥d_{A}^{+}d_{A}^{∗}v∥_{L^{2}(X)} + c (1 + ∥F_{A}^{−}∥_{C^{0}(X)})^{1/2} ∥v∥_{L^{2}(X)} \\ & + ∥F_{A}^{+}∥_{C^{0}(X)}∥v∥_{L^{2}(X)}. \end{aligned} (5.9)

Proof. From the Bochner-Weitzenböck formula for d_{A}d_{A}^{∗} + 2d_{A}^{∗}d_{A}^{+} in (5.6) and integration by parts, we have:

∥∇_{A}a∥_{L^{2}}^{2} = (∇_{A}^{∗}∇_{A}a, a)

= (d_{A}d_{A}^{∗}a, a) + 2 (d_{A}^{∗}d_{A}^{+}a, a) − ({Ric, a}, a) + 2 ({F_{A}^{−}, a}, a)

≤ ∥d_{A}^{∗}a∥_{L^{2}}^{2} + 2∥d_{A}^{+}a∥_{L^{2}}^{2} + c (1 + ∥F_{A}^{−}∥_{C^{0}}) ∥a∥_{L^{2}}^{2}

which gives (5.8). If a = d_{A}^{∗}v, then d_{A}^{∗}d_{A}^{∗}v = (d_{A}^{+}d_{A})^{∗}v = (F_{A}^{+})^{∗}v, so that

∥d_{A}^{∗}d_{A}^{∗}v∥_{L^{2}} ≤ ∥F_{A}^{+}∥_{C^{0}}∥v∥_{L^{2}}.

Thus, (5.9) follows from (5.8) and the above inequality. □

Since d_{A}^{∗} + d_{A}^{+} is an elliptic operator, estimates of the above form follow from the general theory of linear elliptic operators. However, the preceding elementary derivation using the Bochner-Weitzenböck formula gives us a constant whose dependence on the curvature terms F_{A}^{−} and F_{A}^{+} is made explicit. In particular, we see that the estimate is only useful when we have a uniform C^{0} bound on F_{A}^{−} independent of A, which is not possible when A bubbles. At the cost of introducing a slightly stronger norm than the L^{2} norm on the right hand side of the estimate above, we can derive an L_{1,A}^{2} bound for a = d_{A}^{∗}v with an estimate constant depending on ∥F_{A}^{−}∥_{L^{2}(X)} rather than ∥F_{A}^{−}∥_{C^{0}(X)}. Specifically, Equation (5.5) and Lemma 5.5 give the following L_{2,A}^{2} estimates for sections of Λ^{+} ⊗ 𝔤_{E}:

Lemma 5.8. Continue the hypotheses of Lemma 5.7. Then the following estimate holds for any v ∈ L_{2}^{♯,2}(Λ^{+} ⊗ 𝔤_{E}):

∥v∥_{L_{2,A}^{2}(X)} + ∥v∥_{C^{0}(X)} ≤ c(1 + ∥F_{A}∥_{L^{2}(X)})(∥∇_{A}^{∗}∇_{A}v∥_{L^{♯,2}(X)} + ∥v∥_{L^{2}(X)}).

We now replace the covariant Laplacian ∇_{A}^{∗}∇_{A} in the estimates of Lemma 5.8 by the Laplacian d_{A}^{+}d_{A}^{∗} via the Bochner formula (5.7) to give:

Lemma 5.9. Continue the hypotheses of Lemma 5.8. Then there is a positive constant ε = ε(c) such that the following holds. If ∥F_{A}^{+}∥_{L^{♯,2}(X)} < ε, then

∥v∥_{L_{2,A}^{2}(X)} + ∥v∥_{C^{0}(X)} ≤ c(1 + ∥F_{A}∥_{L^{2}(X)})(∥d_{A}^{+}d_{A}^{∗}v∥_{L^{♯,2}(X)} + ∥v∥_{L^{2}(X)}).

Proof. From (5.5) and Lemma 4.3 we have

∥∇_{A}^{∗}∇_{A}v∥_{L^{♯,2}} ≤ 2∥d_{A}^{+}d_{A}^{∗}v∥_{L^{♯,2}} + c∥v∥_{L^{♯,2}} + c∥F_{A}^{+}∥_{L^{♯,2}}∥v∥_{C^{0}}.

Combining the preceding estimate with that of Lemma 5.8, together with the embedding and interpolation inequalities ∥v∥_{L^{♯}} ≤ c∥v∥_{L^{4}} ≤ c∥v∥_{L^{2}}^{1/2}∥v∥_{C^{0}}^{1/2}, and using rearrangement with the last term yields the desired bound. In particular, by choosing ε(c) small enough that c∥F_{A}^{+}∥_{L^{♯,2}}∥v∥_{C^{0}} ≤ 1/2, we may use rearrangement to bring the right-hand term ∥v∥_{C^{0}} to the left-hand side. □

Since ∥d_{A}^{∗}v∥_{L_{1,A}^{2}} ≤ ∥v∥_{L_{2,A}^{2}}, Lemma 5.9 yields an L_{1,A}^{2} estimate for d_{A}^{∗}v:

Corollary 5.10. Continue the hypotheses of Lemma 5.9. Then:

∥d_{A}^{∗}v∥_{L_{1,A}^{2}(X)} ≤ c(1 + ∥F_{A}∥_{L^{2}(X)})(∥d_{A}^{+}d_{A}^{∗}v∥_{L^{♯,2}(X)} + ∥v∥_{L^{2}(X)}).

Note that if a ∈ Ω^{1}(𝔤_{E}) is L^{2}-orthogonal to Ker d_{A}^{+}, so that a = d_{A}^{∗}v for some v ∈ Ω^{+}(𝔤_{E}), and Ker d_{A}^{+}d_{A}^{+,∗} = 0, then the estimate of Corollary 5.10 can be written in the more familiar form

∥a∥_{L_{1,A}^{2}(X)} ≤ c(1 + ∥F_{A}∥_{L^{2}(X)})(∥d_{A}^{+}a∥_{L^{♯,2}(X)} + ν_{2}[A]^{−1/2}∥a∥_{L^{2}(X)}), (5.10)

where we make use of the eigenvalue estimate ∥v∥_{L^{2}} ≤ ν_{2}[A]^{−1/2}∥d_{A}^{∗}v∥_{L^{2}}; the term d_{A}^{+}a above can be replaced by (d_{A}^{+}+d_{A}^{∗})a without changing the estimate constants. Here, ν_{2}[A] is the least positive eigenvalue of the Laplacian d_{A}^{+}d_{A}^{+,∗}.

6. Existence of gauge transformations via the method of continuity.

In this section we complete the proof of Theorem 1.1, and hence the proof of Theorem 1.2, using the method of continuity. The strategy broadly follows that of Uhlenbeck’s proof of Theorem 2.1 in [26]. The main new technical difficulty, not present in [26], is the need to compare distances in the Coulomb-gauge slice S_{A_{0}} ⊂ A_{E}^{k} through the connection A_{0} and gaugeinvariant distances in B_{E}^{k} from the point [A_{0}]. It is at this stage of the method of continuity (in proving openness—see Lemma 6.6)—that we need to employ the special norms and Green’s operator estimates described in Sections 4 and 5 in order to achieve the requisite C^{0} control of gauge transformations; the proof of closedness works, as one would expect, with standard Sobolev L^{4} and L_{1}^{2} norms. In [26], the L^{2} norm of the curvature F_{A} essentially serves as a gauge-invariant L_{1}^{2} measure of distance from [A] to [Γ], where Γ is the product connection on the product G bundle over the unit ball. Our goal in this section is to prove:

Theorem 6.1. Let X be a closed, smooth four-manifold with metric g and let G be a compact Lie group. Then there are positive constants c, z with the following significance. Let E be a G bundle over X and suppose that k ≥ 2 is an integer. Given a point [A_{0}] in B_{E}^{k}, let ν_{0}[A_{0}] be the least positive eigenvalue of the Laplacian ∇_{A_{0}}^{∗}∇_{A_{0}} on Ω^{0}(𝔤_{E}) and set K_{0} = (1 + ν_{0}[A_{0}]^{−1})(1 + ∥F_{A_{0}}∥_{L^{2}}). Let ε_{1} be a constant satisfying 0 < ε_{1} ≤ zK_{0}^{−2}(1 + ν_{0}[A_{0}]^{−1/2})^{−1}. Then the following hold:

(1) For any [A] ∈ B_{E}^{k} with dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) < ε_{1}, then u ∈ G_{E}^{k+1} exists

such that

(a) d_{A_{0}}^{∗}(u(A) − A_{0}) = 0,

(b) ∥u(A) − A_{0}∥_{L^{2♯,4}} ≤ cK_{0} dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]).

(2) For any [A] ∈ B_{E}^{k} with dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) < ε_{1}, then u ∈ G_{E}^{k+1} exists

such that

(a) d_{A_{0}}^{∗}(u(A) − A_{0}) = 0,

(b) ∥u(A) − A_{0}∥_{L^{2♯,4}} ≤ cK_{0} dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]),

(c) ∥u(A) − A_{0}∥_{L_{1,A_{0}}^{2}} ≤ cK_{0} dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]).

Our first proof of Theorem 6.1, via the method of continuity, occupies Sections 6.1, 6.2 and 6.3. A rather different proof, via a direct application of the inverse function theorem using L_{2}^{♯,2} gauge transformations, is given in Section 8.

6.1. Distance functions on the quotient space. Our first task is to verify the existence of minimizing gauge transformations u ∈ G_{E}^{k+1} for the family of distance functions on B_{E}^{k} defined above: This is established in Lemma 6.3 and the proof uses the following version of Uhlenbeck’s weak compactness theorem.

Proposition 6.2. Let X be a closed, smooth, Riemannian four-manifold, let G be a compact Lie group, let M be a positive constant, let A_{0} be an L_{2}^{2} connection on a G bundle E over X. If {A_{α}} is a sequence of L_{2}^{2} connections on E such that ∥F_{A_{α}}∥_{L_{1,A_{0}}^{2}} ≤ M, then there is a subsequence {α′} ⊂ {α} and

a sequence of L_{3}^{2} gauge transformations {u_{α′}} such that u_{α′}(A_{α′}) converges weakly in L_{2,A_{0}}^{2} and strongly in L_{1,A_{0}}^{p}, for 1 ≤ p < 4, to an L_{2}^{2} connection A_{∞} on E.

Proof. From the Sobolev embedding L_{1}^{2} ⊂ L^{p}, 2 < p < 4, we obtain a uniform L^{p} bound ∥F_{A_{α}}∥_{L^{p}} ≤ cM and so, according to [26, Theorem 3.6], there is a subsequence {α′} ⊂ {α} and a sequence of L_{2}^{p} gauge transformations {u_{α′}} such that u_{α′}(A_{α′}) converges weakly in L_{1,A_{0}}^{p} to an L_{1}^{p} connection A_{∞} on E. The stronger conclusion above is obtained simply by reworking the proof of Theorem 3.6 in [26], using the following local estimate for the connections A_{α} over small balls B ⊂ X. Theorem 2.1 of [26] provides a sequence of local trivializations v_{α} : P |_{B} → B × G such that a_{α} = v_{α}(A_{α}) − Γ satisfies d^{∗}a_{α} = 0 and

∥a_{α}∥_{L_{1}^{p}(B)} ≤ c∥F_{A_{α}}∥_{L^{p}(B)}, 2 ≤ p < 4,

where Γ is the product connection. Now F_{A_{α}} = da_{α} + a_{α} ∧ a_{α}, so

∥a_{α}∥_{L_{2}^{2}(B)} ≤ ∥da_{α}∥_{L_{1}^{2}(B)} + ∥a_{α}∥_{L_{1}^{2}(B)}
≤ ∥a_{α} ∧ a_{α}∥_{L_{1}^{2}(B)} + ∥F_{A_{α}}∥_{L_{1}^{2}(B)} + ∥a_{α}∥_{L_{1}^{2}(B)}.

Now, using the multiplication L^{6} × L^{3} → L^{2}, the embeddings L_{1}^{3} ⊂ L_{1}^{12/5} ⊂ L^{6} and d(a_{α} ∧ a_{α}) = da_{α} ∧ a_{α} − a_{α} ∧ da_{α}, we have

∥d(a_{α} ∧ a_{α})∥_{L^{2}} ≤ c∥da_{α}∥_{L^{3}}∥a_{α}∥_{L^{6}} ≤ c∥a_{α}∥_{L_{1}^{3}}^{2},

while ∥a_{α} ∧ a_{α}∥_{L^{2}} ≤ ∥a_{α}∥_{L^{4}(B)}^{2} ≤ c∥a_{α}∥_{L_{1}^{2}(B)}^{2}. Hence, we obtain

∥a_{α}∥_{L_{2}^{2}(B)} ≤ c∥F_{A_{α}}∥_{L_{1}^{2}(B)} + c∥a_{α}∥_{L_{1}^{3}(B)}^{2} + ∥a_{α}∥_{L_{1}^{2}(B)}
≤ c∥F_{A_{α}}∥_{L_{1}^{2}(B)}(1 + ∥F_{A_{α}}∥_{L_{1}^{2}(B)})
≤ c∥F_{A_{α}}∥_{L_{1,A_{0}}^{2}(B)}(1 + ∥F_{A_{α}}∥_{L_{1,A_{0}}^{2}(B)})(1 + ∥A_{0} − Γ∥_{L_{1}^{2}}).

In particular, the sequence of Coulomb-gauge, local connection matrices {a_{α}} is bounded in L_{2}^{2}(B), so we can extract a weakly L_{2}^{2}(B)-convergent and strongly L_{1}^{p}(B)-convergent subsequence, via the compactness of embedding L_{2}^{2}(B) ⊂ L_{1}^{p}(B) when 1 ≤ p < 4. The patching argument used to complete the proof of Uhlenbeck’s theorem now proceeds exactly as in [26] to give the desired conclusion. □

The proposition is used to extract the desired convergence in the next lemma.

Lemma 6.3. For any points [A_{0}], [A] in B_{E}^{k} there are gauge transformations such that the following equalities hold:

(1)  dist_{L^{4}}([A], [A_{0}]) = ∥u(A) − A_{0}∥_{L^{4}(X)}, u ∈ G_{E}^{3}
(2) dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) = ∥v(A) − A_{0}∥_{L_{1,A_{0}}^{♯,2}} , v ∈ G_{E}^{3},
(3) dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) = ∥w(A) − A_{0}∥_{L_{1,A_{0}}^{♯,2}} , w ∈ G_{E}^{3},
(4) dist_{L_{ℓ,A_{0}}^{2}} ([A], [A_{0}]) = ∥w_{ℓ}(A) − A_{0}∥_{L_{ℓ,A_{0}}^{2}} ,  ℓ = 1 and 3 ≤ ℓ ≤ k,

where w_{1} ∈ G_{E}^{3} and w_{ℓ} ∈ G_{E}^{ℓ+1} in (4). The above distance functions (including the ℓ = 2 distance function in (4)) are continuous with respect to the quotient L_{k}^{2} topology on B_{E}^{k}.

Proof. Consider (1). Let {u_{α}} be a minimizing sequence in G_{E}^{k+1}, so ∥u_{α}(A)− A_{0}∥_{L^{4}} converges to dist_{L^{4}}([A], [A_{0}]) as α → ∞. Setting B_{α} = u_{α}(A) = A − (d_{A}u_{α})u_{α}^{−1} ∈ A_{E}^{k}, we see that B_{α}u_{α} = Au_{α} − d_{A}u_{α} = Au_{α} − d_{A_{0}}u_{α} − [A − A_{0}, u_{α}], and thus

d_{A_{0}}u_{α} = u_{α}(A − A_{0}) − (B_{α} − A_{0})u_{α}. (6.1)

Therefore, as ∥u_{α}∥_{C^{0}} ≤ c(G), we have

∥∇_{A_{0}}u_{α}∥_{L^{4}} ≤ c(∥A − A_{0}∥_{L^{4}} + ∥B_{α} − A_{0}∥_{L^{4}}),

so the sequence {u_{α}} ⊂ L_{k+1}^{2}(𝔤𝔩(E)) is bounded in L_{1,A_{0}}^{4}(𝔤𝔩(E)). So, passing to a subsequence, we may suppose that {u_{α}} converges weakly in L_{1,A_{0}}^{4}(𝔤𝔩(E)) and strongly in L^{q}(𝔤𝔩(E)), via the compact embedding L_{1}^{4} ⊂ L^{q}, for any 1 ≤ q < ∞, to a limit u ∈ L_{1}^{4}(𝔤𝔩(E)).

We also have F_{B_{α}} = F_{u_{α}(A)} = u_{α}F_{A}u_{α}^{−1}, so ∥F_{B_{α}}∥_{L^{2}} = ∥F_{A}∥_{L^{2}} and as

∇_{A_{0}}F_{B_{α}} = (∇_{A_{0}}u_{α}) ⊗ F_{A}u_{α}^{−1} + u_{α}(∇_{A_{0}}F_{A})u_{α}^{−1} − u_{α}F_{A} ⊗ u_{α}^{−1}(∇_{A_{0}}u_{α})u_{α}^{−1},

we see that

\begin{aligned}∥∇_{A_{0}}F_{B_{α}}∥_{L^{2}} & ≤ c(∥∇_{A_{0}}u_{α}∥_{L^{4}}∥F_{A}∥_{L^{4}} + ∥∇_{A_{0}}F_{A}∥_{L^{2}}) \\ & ≤ c(1 + ∥u_{α}∥_{L_{1,A_{0}}^{4}} )∥F_{A}∥_{L_{1,A_{0}}^{2}} . \end{aligned} (6.2)

Hence, the sequence of L_{k}^{2} connections {B_{α}} has curvature uniformly bounded in L_{1,A_{0}}^{2}: Proposition 6.2 implies, after passing to a subsequence, that the sequence {B_{α}} converges weakly in L_{2,A_{0}}^{2} and strongly in L_{1,A_{0}}^{p}, for 1 ≤ p < 4, to an L_{2}^{2} G connection B on E. From (6.1) we obtain

d_{A_{0}}u = u(A − A_{0}) − (B − A_{0})u, (6.3)

a first-order linear elliptic equation in u with L_{2}^{2} coefficients. Therefore, u ∈ L_{3}^{2}(𝔤𝔩(E)) and B = u(A) = A − (d_{A}u)u^{−1} lies in A_{E}^{2}. It is not a priori clear that the limit u actually lies in G_{E}^{3} (since the convergence was only weakly L_{1,A_{0}}^{4}(𝔤𝔩(E)) and strongly L^{q}(𝔤𝔩(E))): However, the argument of the last paragraph in the proof of Lemma 4.2.4 in [5, p. 130] applies (using the compactness of the structure group G) and shows that the limit gauge transformation u lies in G_{E}^{3}. Since B_{α} = u_{α}(A) converges strongly in L_{1,A_{0}}^{p} to u(A) we now have

dist_{L^{4}}([A], [A_{0}]) = lim_{α→∞} ∥u_{α}(A) − A_{0}∥_{L^{4}} = ∥u(A) − A_{0}∥_{L^{4}},

as required in (1). The same argument proves Assertions (2) and (3) and Assertion (4) when ℓ = 1. The case ℓ ≥ 3 in (4) is straightforward as we can now apply Lemma 3.1 to obtain the desired convergence.

It remains to check L_{k}^{2} continuity. We just consider (1), as the remaining cases are identical. If [A_{α}] ∈ B_{E}^{k} is a sequence converging to [A_{∞}] ∈ B_{E}^{k}, then there is a sequence of gauge transformations s_{α} ∈ G_{E}^{k+1} such that s_{α}(A_{α}) converges in L_{k,A_{0}}^{2} to A_{∞} ∈ A_{E}^{k} and, in particular, in L^{4}. But then

| dist_{L^{4}}([A_{α}], [A_{0}]) − dist_{L^{4}}([A_{∞}], [A_{0}])|
= | dist_{L^{4}}([s_{α}(A_{α})], [A_{0}]) − dist_{L^{4}}([A_{∞}], [A_{0}])|
≤ dist_{L^{4}}([s_{α}(A_{α})], [A_{∞}]) ≤ ∥s_{α}(A_{α}) − A_{∞}∥_{L^{4}},

and so

lim_{α→∞} dist_{L^{4}}([A_{α}], [A_{0}]) = dist_{L^{4}}([A_{∞}], [A_{0}]),

as desired. □

6.2. Closedness. Let 𝔅 ⊂ B̄_{[A_{0}]}^{1,♯,2}(ε) be the subset of points [A] such that

there exists a gauge transformation u ∈ G_{E}^{k+1} satisfying the conclusions of Assertion (2) of Theorem 6.1; let 𝔅^{∗} ⊂ B̄_{[A_{0}]}^{1,∗,2}(ε) be the subset of points

[A] such that there exists a gauge transformation u ∈ G_{E}^{k+1} satisfying the conclusions of Assertion (1). As in the proof of Theorem 2.1 in [26], we apply the method of continuity to show that 𝔅^{∗} = B̄_{[A_{0}]}^{1,∗,2}(ε) and 𝔅 = B̄_{[A_{0}]}^{1,♯,2}(ε) for small enough ε. Not surprisingly, we have:

Lemma 6.4. The balls B̄_{[A_{0}]}^{1,∗,2}(ε) and B̄_{[A_{0}]}^{1,♯,2}(ε) are connected.

Proof. If [A] ∈ B̄_{[A_{0}]}^{1,♯,2}(ε), there is a gauge transformation u ∈ G_{E}^{k+1} such that ∥u(A) − A_{0}∥_{L_{1,A_{0}}^{♯,2}} ≤ ε. Then A_{t} = A_{0} + t(u(A) − A_{0}), t ∈ [0, 1], is a path in

A_{E}^{k} joining A_{0} to u(A) and ∥A_{t} − A_{0}∥_{L_{1,A_{0}}^{♯,2}} = t∥u(A) − A_{0}∥_{L_{1,A_{0}}^{♯,2}} ≤ tε, so the

path [A_{t}] lies in B̄_{[A_{0}]}^{1,♯,2}(ε) and joins [A_{0}] to [A]. Similarly for B̄_{[A_{0}]}^{1,∗,2}(ε). □

Our task then reduces to showing that 𝔅^{∗} is an open and closed subspace of B̄_{[A_{0}]}^{1,∗,2}(ε) and that 𝔅 is an open and closed subspace of B̄_{[A_{0}]}^{1,♯,2}(ε). First we consider closedness:

CRITICAL-EXPONENT SOBOLEV NORMS AND THE SLICE THEOREM 101

Lemma 6.5. The subspaces 𝔅^{∗} ⊂ B̄_{[A_{0}]}^{1,∗,2}(ε) and 𝔅 ⊂ B̄_{[A_{0}]}^{1,♯,2}(ε) are closed.

Proof. It suffices to consider the second assertion as the same argument yields the first. Suppose [A_{α}] is a sequence of points in 𝔅 which converges in B_{E}^{k} to a point [B_{∞}]. We may suppose, without loss of generality, that A_{α} ∈ A_{E}^{k} is the corresponding sequence of connections, representing the gauge-equivalence classes [A_{α}], which satisfy the defining conditions for 𝔅:

d_{A_{0}}^{∗}(A_{α} − A_{0}) = 0,
∥A_{α} − A_{0}∥_{L^{2♯,4}} ≤ cK_{0} dist_{L_{1,A_{0}}^{♯,2}} ([A_{α}], [A_{0}]),
∥A_{α} − A_{0}∥_{L_{1,A_{0}}^{2}} ≤ cK_{0} dist_{L_{1,A_{0}}^{♯,2}} ([A_{α}], [A_{0}]).

(6.4)

Since [A_{α}] converges in B_{E}^{k} to [B_{∞}], there is a sequence of gauge transformations u_{α} ∈ G_{E}^{k+1} such that B_{α} := u_{α}(A_{α}) converges in L_{k,A_{0}}^{2} to B_{∞} ∈ A_{E}^{k}. Since B_{α} = u_{α}(A_{α}) and d_{A_{0}}^{∗}(A_{α} − A_{0}) = 0, we have

(6.5)  d_{A_{0}}u_{α} = u_{α}(A_{α} − A_{0}) − (B_{α} − A_{0})u_{α},
(6.6)  d_{A_{0}}^{∗}d_{A_{0}}u_{α} = − ∗ (d_{A_{0}}u_{α} ∧ ∗(A_{α} − A_{0})) − (d_{A_{0}}^{∗}(B_{α} − A_{0}))u_{α}
− ∗(∗(B_{α} − A_{0}) ∧ d_{A_{0}}u_{α}),

and so, as ∥u_{α}∥_{C^{0}} ≤ 1,

∥d_{A_{0}}u_{α}∥_{L^{2♯,4}} ≤ ∥A_{α} − A_{0}∥_{L^{2♯,4}} + ∥B_{α} − A_{0}∥_{L^{2♯,4}}
∥d_{A_{0}}^{∗}d_{A_{0}}u_{α}∥_{L^{♯,2}} ≤ ∥d_{A_{0}}u_{α}∥_{L^{2♯,4}}∥A_{α} − A_{0}∥_{L^{2♯,4}} + ∥d_{A_{0}}^{∗}B_{α} − A_{0}∥_{L^{♯,2}}
∥B_{α} − A_{0}∥_{L^{2♯,4}}∥d_{A_{0}}u_{α}∥_{L^{2♯,4}}.

Therefore, the sequence u_{α} is bounded in L_{2,A_{0}}^{2}(𝔤𝔩(E)) and so, passing to a subsequence, we may suppose that u_{α} converges weakly in L_{2,A_{0}}^{2}(𝔤𝔩(E)) (and strongly in L_{1,A_{0}}^{p}, for any p < 4 via the compact embedding L_{2}^{2} ⊂ L_{1}^{p}) to a limit u_{∞} ∈ L^{∞} ∩ L_{2,A_{0}}^{2}(𝔤𝔩(E)).

On the other hand, using A_{α} = u_{α}^{−1}(B_{α}), we have ∥F_{A_{α}}∥_{L^{2}} = ∥F_{B_{α}}∥_{L^{2}} and the derivation of (6.2) gives

∥∇_{A_{0}}F_{A_{α}}∥_{L^{2}} ≤ c(1 + ∥u_{α}∥_{L_{1,A_{0}}^{4}} )∥F_{B_{α}}∥_{L_{1,A_{0}}^{2}} ,

so the sequence A_{α} has curvature uniformly bounded in L_{1,A_{0}}^{2}. Thus, after passing to a subsequence we may assume by Proposition 6.2 that the sequence A_{α} converges weakly in L_{2,A_{0}}^{2} and strongly in L_{1,A_{0}}^{p}, 2 ≤ p < 4, to a limit A_{∞} ∈ A_{E}^{2}.

Taking weak limits in (6.5) and (6.6) yields

d_{A_{0}}u_{∞} = u_{∞}(A_{∞} − A_{0}) − (B_{∞} − A_{0})u_{∞}. (6.7)

The equation (6.7) is first order, linear, elliptic in u_{∞} ∈ L^{∞} ∩ L_{2}^{2} with L_{2}^{2} coefficients. Hence, u_{∞} is in L_{3}^{2}(𝔤𝔩(E)) and in particular, in G_{E}^{3}, while B_{∞} = u_{∞}(A_{∞}). From (6.7) we see that

A_{∞} − A_{0} = u_{∞}^{−1}(B_{∞} − A_{0})u_{∞} + u_{∞}^{−1}d_{A_{0}}u_{∞}

and so, as d_{A_{0}}^{∗}(A_{∞} − A_{0}) = 0, we have

d_{A_{0}}^{∗}(u_{∞}^{−1}d_{A_{0}}u_{∞} + u_{∞}^{−1}(B_{∞} − A_{0})u_{∞}) = 0. (6.8)

This is a second-order elliptic equation for u_{∞} ∈ G_{E}^{3} with L_{k}^{2} coefficients: In particular, since u_{∞} ∈ L_{2}^{p} for 2 ≤ p ≤ 4, a standard elliptic bootstrapping argument then implies that u_{∞} ∈ L_{k+1}^{2} (see, for example, the proof of Proposition 3.3 in [6]) and therefore A_{∞} = u_{∞}^{−1}(B_{∞}) ∈ A_{E}^{k}.

Now, taking weak limits in (6.4), we have

d_{A_{0}}^{∗}(A_{∞} − A_{0}) = lim_{α→∞} d_{A_{0}}^{∗}(A_{α} − A_{0}) = 0,
∥A_{∞} − A_{0}∥_{L^{2♯,4}} = lim_{α→∞} ∥A_{α} − A_{0}∥_{L^{2♯,4}} ≤ lim_{α→∞} cK_{0} dist_{L_{1,A_{0}}^{♯,2}} ([A_{α}], [A_{0}]),
∥A_{α} − A_{0}∥_{L_{1,A_{0}}^{♯,2}} = lim_{α→∞} ∥A_{α} − A_{0}∥_{L_{1,A_{0}}^{♯,2}} ≤ lim_{α→∞} cK_{0} dist_{L_{1,A_{0}}^{♯,2}} ([A_{α}], [A_{0}]).

Moreover, as B_{∞} = u_{∞}(A_{∞}) and u_{∞} ∈ G_{E}^{k+1},

lim_{α→∞} dist_{L_{1,A_{0}}^{♯,2}} ([A_{α}], [A_{0}]) = dist_{L_{1,A_{0}}^{♯,2}} ([B_{∞}], [A_{0}]) = dist_{L_{1,A_{0}}^{♯,2}} ([A_{∞}], [A_{0}]),
lim_{α→∞} dist_{L_{1,A_{0}}^{♯,2}} ([A_{α}], [A_{0}]) = dist_{L_{1,A_{0}}^{♯,2}} ([B_{∞}], [A_{0}]) = dist_{L_{1,A_{0}}^{♯,2}} ([A_{∞}], [A_{0}]),

where the L_{k}^{2} continuity of the distance functions is given by Lemma 6.3. Therefore, [B_{∞}] = [A_{∞}] ∈ 𝔅. Thus, 𝔅 is closed in B_{E}^{k} and in particular, closed in B̄_{[A_{0}]}^{1,♯,2}(ε), as desired. □ 6.3. Openness. We must first compare distances from the connection A_{0} in the Coulomb slice through A_{0} in A_{E}^{k} and gauge-invariant distances in B_{E}^{k} from the point [A_{0}]:

Lemma 6.6. Let (X, g) be a closed, smooth, Riemannian four-manifold. Then there are positive constants c, z with the following significance. Let K_{0} = (1 + ν_{0}[A_{0}]^{−1})(1 + ∥F_{A_{0}}∥_{L^{2}}). If A ∈ A_{E}^{k} satisfies

• d_{A_{0}}^{∗}(A − A_{0}) = 0,

• ∥A − A_{0}∥_{L^{2♯,4}} ≤ zK_{0}^{−1}, then the following hold:

(1) If dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) ≤ zK_{0}^{−1}, then

∥A − A_{0}∥_{L^{2♯,4}} ≤ cK_{0} dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]);

(2) If dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) ≤ zK_{0}^{−1}, then

∥A − A_{0}∥_{L^{2♯,4}} ≤ cK_{0} dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]),
∥A − A_{0}∥_{L_{1,A_{0}}^{2}} ≤ cK_{0} dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]).

Proof. Recall that for either distance function, minimizing gauge transformations in G_{E}^{3} exist by Lemma 6.3; for convenience, we denote both by u ∈ G_{E}^{3} although they need not a priori coincide. Since B := u(A) = A − (d_{A}u)u^{−1} ∈ A_{E}^{2}, we have

u(A) − A_{0} = u(A − A_{0})u^{−1} − (d_{A_{0}}u)u^{−1}.

Our task, in essence, is to estimate the second term on the right above. Rewriting this equality gives a first-order, linear elliptic equation in u with L_{2}^{2} coefficients:

d_{A_{0}}u = u(A − A_{0}) − (B − A_{0})u. (6.9)

Let u_{0} ∈ L_{3}^{2}(𝔤𝔩(E)) be the L^{2} orthogonal projection of u ∈ G_{E}^{3} ⊂ L_{3}^{2}(𝔤𝔩(E)) onto Ker(d_{A_{0}}|_{L_{3}^{2}})^{⊥}, so u = u_{0} + γ, where γ ∈ Ker d_{A_{0}} ⊂ Ω^{0}(𝔤𝔩(E)). Thus, as d_{A_{0}}^{∗}(A − A_{0}) = 0 and d_{A_{0}}u = d_{A_{0}}u_{0}, we see that

d_{A_{0}}^{∗}d_{A_{0}}u_{0} = − ∗ (d_{A_{0}}u ∧ ∗(A − A_{0})) + ud_{A_{0}}^{∗}(A − A_{0})
− (d_{A_{0}}^{∗}(B − A_{0}))u − ∗(∗(B − A_{0}) ∧ d_{A_{0}}u)
= − ∗ (d_{A_{0}}u_{0} ∧ ∗(A − A_{0})) − (d_{A_{0}}^{∗}(B − A_{0}))u
− ∗(∗(B − A_{0}) ∧ d_{A_{0}}u_{0}).

Therefore, using the bound ∥u∥_{C^{0}} ≤ 1 for any u ∈ G_{E}^{3} (as the representation for G is orthogonal), we have

∥Δ_{A_{0}}^{0}u_{0}∥_{L^{♯,2}} ≤ ∥d_{A_{0}}u_{0}∥_{L^{2♯,4}}∥A − A_{0}∥_{L^{2♯,4}} + ∥d_{A_{0}}^{∗}(B − A_{0})∥_{L^{♯,2}}∥u∥_{C^{0}}
+ ∥B − A_{0}∥_{L^{2♯,4}}∥d_{A_{0}}u_{0}∥_{L^{2♯,4}}
≤ C (∥A − A_{0}∥_{L^{2♯,4}} + ∥B − A_{0}∥_{L^{2♯,4}}) ∥d_{A_{0}}^{∗}d_{A_{0}}u_{0}∥_{L^{♯,2}}
+ ∥d_{A_{0}}^{∗}(B − A_{0})∥_{L^{♯,2}},

where C = cK_{0}. Now ∥B − A_{0}∥_{L^{2♯,4}} ≤ c∥B − A_{0}∥_{L_{1,A_{0}}^{2}} via the embed-

ding L_{1}^{2} ⊂ L^{2♯,4} of Lemma 4.1. For either dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) ≤ \frac{1}{4}C^{−1} or

dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) ≤ \frac{1}{4}C^{−1} and ∥A−A_{0}∥_{L^{2♯,4}} ≤ \frac{1}{4}C^{−1}, rearrangement yields

∥Δ_{A_{0}}^{0}u_{0}∥_{L^{♯,2}} ≤ 2∥d_{A_{0}}^{∗}(B − A_{0})∥_{L^{♯,2}}. (6.10)

On the other hand, from Lemma 5.6 we have

\begin{aligned}∥u_{0}∥_{L_{2,A_{0}}^{♯,2}} & ≤ C∥Δ_{A_{0}}^{0}u_{0}∥_{L^{♯,2}}, \\ ∥u_{0}∥_{L_{1,A_{0}}^{4}} & ≤ C∥Δ_{A_{0}}^{0}u_{0}∥_{L^{♯,2}}, \end{aligned} (6.11)

where C = cK_{0} and the second bound follows from the embedding L_{2}^{2} ⊂ L_{1}^{4}. So, combining (6.10) and (6.11) yields:

\begin{aligned}∥u_{0}∥_{L_{2,A_{0}}^{♯,2}} & ≤ C∥d_{A_{0}}^{∗}(B − A_{0})∥_{L^{♯,2}}, \\ ∥u_{0}∥_{L_{1,A_{0}}^{4}} & ≤ C∥d_{A_{0}}^{∗}(B − A_{0})∥_{L^{♯,2}}. \end{aligned} (6.12)

Consequently, using d_{A_{0}}u = d_{A_{0}}u_{0} and (6.9) rewritten in the form,

u^{−1}(B − A_{0})u = (A − A_{0}) − u^{−1}d_{A_{0}}u_{0}, (6.13)

we obtain

(6.14)  ∥A − A_{0}∥_{L^{2♯,4}} ≤ ∥u^{−1}(B − A_{0})u∥_{L^{2♯,4}} + ∥u^{−1}d_{A_{0}}u_{0}∥_{L^{2♯,4}},
(6.15)  ∥A − A_{0}∥_{L_{1,A_{0}}^{2}} ≤ ∥u^{−1}(B − A_{0})u∥_{L_{1,A_{0}}^{2}} + ∥u^{−1}d_{A_{0}}u_{0}∥_{L_{1,A_{0}}^{2}} .

From (6.14) and (6.12), we see that

∥A − A_{0}∥_{L^{2♯,4}} ≤ ∥B − A_{0}∥_{L^{2♯,4}} + ∥d_{A_{0}}u_{0}∥_{L^{2♯,4}}
≤ dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) + C∥d_{A_{0}}^{∗}(B − A_{0})∥_{L^{♯,2}}
≤ (1 + C) dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]),

giving the desired L^{2♯,4} estimate for A − A_{0}.

Considering the first term in (6.13), we have

∇_{A_{0}}(u^{−1}(B − A_{0})u) = −u^{−1}(∇_{A_{0}}u)u^{−1} ⊗ (B − A_{0})u

+ u^{−1}(∇_{A_{0}}(B − A_{0}))u + u^{−1}(B − A_{0}) ⊗ ∇_{A_{0}}u,

and so applying (6.12), noting that ∇_{A_{0}}u = ∇_{A_{0}}u_{0} and ∥u∥_{C^{0}} ≤ 1, we have

∥∇_{A_{0}}(u^{−1}(B − A_{0})u)∥_{L^{2}} ≤ ∥∇_{A_{0}}u_{0}∥_{L^{4}}∥B − A_{0}∥_{L^{4}} + ∥∇_{A_{0}}(B − A_{0})∥_{L^{2}}

≤ C dist_{L_{1,A_{0}}^{♯,2}}^{2} ([A], [A_{0}]) + dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]).

Thus, if dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) ≤ \frac{1}{4}C^{−1}, say, we obtain

∥∇_{A_{0}}(u^{−1}(B − A_{0})u)∥_{L^{2}} ≤ 2 dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]). (6.16)

Similarly, considering the second term in (6.13), we have

∇_{A_{0}}(u^{−1}d_{A_{0}}u_{0}) = −u^{−1}(∇_{A_{0}}u)u^{−1} ⊗ d_{A_{0}}u + u^{−1}∇_{A_{0}}d_{A_{0}}u

and therefore, by (6.12), we see that

∥∇_{A_{0}}(u^{−1}d_{A_{0}}u_{0})∥_{L^{2}} ≤ ∥∇_{A_{0}}u_{0}∥_{L^{4}}^{2} + ∥∇_{A_{0}}^{2}u_{0}∥_{L^{2}}

≤ C dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) (1 + C dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}])) .

Provided dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) ≤ \frac{1}{4}C^{−1}, we obtain

∥∇_{A_{0}}(u^{−1}d_{A_{0}}u_{0})∥_{L^{2}} ≤ 2C dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]). (6.17)

Taking the L^{2} norm of (6.13) and applying (6.12) to estimate the second term gives

∥A − A_{0}∥_{L^{2}} ≤ ∥B − A_{0}∥_{L^{2}} + ∥d_{A_{0}}u_{0}∥_{L^{2}}
≤ dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) + C∥d_{A_{0}}^{∗}(B − A_{0})∥_{L^{♯,2}},

and so

∥A − A_{0}∥_{L^{2}} ≤ (1 + C) dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]). (6.18)

Combining the estimates (6.15), (6.16), (6.17), and (6.18) yields

∥A − A_{0}∥_{L_{1,A_{0}}^{2}} ≤ 2(1 + C) dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]),

giving the desired L_{1,A_{0}}^{2} estimate for A − A_{0}. □

Naturally, a comparison—going in the other direction—of distances from A_{0} in the Coulomb slice in A_{E}^{k} through A_{0} and gauge-invariant distances in B_{E}^{k} from the point [A_{0}] is elementary: If A ∈ S_{A_{0}} and ∥A − A_{0}∥_{L_{k,A_{0}}^{2}} < δ, say, then Lemma 4.1 implies that

dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) ≤ c∥A − A_{0}∥_{L_{k,A_{0}}^{2}} < cδ,  k ≥ 1,
dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) ≤ c∥A − A_{0}∥_{L_{k,A_{0}}^{2}} < cδ,  k ≥ 2, (6.19)

for some positive constant c(X, g, k). The observation is used in concluding that 𝔅^{∗}, 𝔅 are open subspaces of B̄_{[A_{0}]}^{1,∗,2}(ε_{1}), B̄_{[A_{0}]}^{1,♯,2}(ε_{1}), respectively:

Lemma 6.7. Let (X, g) be a closed, smooth, Riemannian four-manifold and let G be a compact Lie group. Then there is a positive constant z with the following significance. Let K_{0} = (1 + ν_{0}[A_{0}]^{−1})(1 + ∥F_{A_{0}}∥_{L^{2}}). If ε_{1} < zK_{0}^{−2}(1 + ν_{0}[A_{0}]^{−1/2})^{−1}, then:

• 𝔅^{∗} is an open subspace of B̄_{[A_{0}]}^{1,∗,2}(ε_{1});

• 𝔅 is an open subspace of B̄_{[A_{0}]}^{1,♯,2}(ε_{1}).

Proof. It suffices to consider the second assertion, as the argument for the first is identical. Suppose [A] ∈ 𝔅 and that A ∈ A_{E}^{k} is a representative satisfying the defining conditions for 𝔅. Then A satisfies d_{A_{0}}^{∗}(A − A_{0}) = 0 and the estimates

∥A − A_{0}∥_{L^{2♯,4}} ≤ c_{0}K_{0} dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]), ≤ c_{0}K_{0}ε_{1}
∥A − A_{0}∥_{L_{1,A_{0}}^{2}} ≤ c_{0}K_{0} dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) ≤ c_{0}K_{0}ε_{1},

while ∥A − A_{0}∥_{L^{2♯,4}} ≤ c_{1}∥A − A_{0}∥_{L_{1,A_{0}}^{2}} via the Sobolev embedding L_{1}^{2} ⊂ L^{2♯,4} and Kato’s inequality. Consequently, if c_{1}c_{0}K_{0}ε_{1} ≤ \frac{1}{2}ε_{0}, then A ∈ B_{A_{0}}^{4}(ε_{0}) ⊂ A_{E}^{k} and we see that

B̄_{[A_{0}]}^{1,♯,2}(ε_{1}) ⊂ π (B_{[A_{0}]}^{4}(ε_{0})) .

Lemma 3.6 implies that the map π : B_{[A_{0}]}^{4}(ε_{0})/ Stab_{A_{0}} → B_{E}^{k} given by A′ ↦ [A′] is a local homeomorphism onto its image π(B_{[A_{0}]}^{4}(ε_{0})) for any ε_{0} < z(1 + ν_{0}[A_{0}]^{−1/2})^{−1}. In particular, if A′ ∈ B_{[A_{0}]}^{4}(ε_{0}) and ∥A′ − A∥_{L_{k,A_{0}}^{2}} < δ,

then A′ ∈ B_{[A_{0}]}^{1,♯,2}(ε_{1}) ⊂ B̄_{[A_{0}]}^{1,♯,2}(ε_{1}) for small enough δ.

The embedding L_{1}^{2} ⊂ L^{2♯,4} and Lemma 6.6 imply that if ∥A′ −A_{0}∥_{L_{1,A_{0}}^{2}} ≤ zK_{0}^{−1} and dist_{L_{1,A_{0}}^{♯,2}} ([A′], [A_{0}]) ≤ zK_{0}^{−1}, then

∥A′ − A_{0}∥_{L^{2♯,4}} ≤ cK_{0} dist_{L_{1,A_{0}}^{♯,2}} ([A′], [A_{0}]) ≤ cK_{0}ε_{1},
∥A′ − A_{0}∥_{L_{1,A_{0}}^{2}} ≤ cK_{0} dist_{L_{1,A_{0}}^{♯,2}} ([A′], [A_{0}]) ≤ cK_{0}ε_{1}.

These inequalities are satisfied by A; moreover dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) ≤ ε_{1} and

∥A − A_{0}∥_{L_{1,A_{0}}^{2}} ≤ c_{0}Kε_{1}. Require that ε_{1} ≤ \frac{1}{2}zK_{0}^{−1} and c_{0}K_{0}ε_{1} ≤ \frac{1}{2}zK_{0}^{−1}, so ε_{1} ≤ \frac{1}{2}z min\{1, c_{0}\}K_{0}^{−2}. Hence, if A′ is L_{k,A_{0}}^{2}-close enough to A (where k ≥ 2), we can ensure [A′] obeys the last two defining conditions for 𝔅 and so [A′] ∈ 𝔅. Thus, 𝔅 ⊂ B̄_{[A_{0}]}^{1,♯,2}(ε_{1}) is open, as desired. □

We can now conclude the proof of Theorem 6.1:

Proof of Theorem 6.1. Lemmas 6.5 and 6.7 imply that 𝔅 is an open and closed subset of the connected space B̄_{[A_{0}]}^{1,♯,2}(ε_{1}), so 𝔅 = B̄_{[A_{0}]}^{1,♯,2}(ε_{1}). Similarly for 𝔅^{∗} and B̄_{[A_{0}]}^{1,∗,2}(ε_{1}) and hence the result follows. □

Similarly, we conclude the proofs of our main theorems:

Proof of Theorem 1.1. Given Theorem 6.1, the only assertion left unaccounted for is the uniqueness of the gauge transformation u ∈ G_{E}^{k+1}, modulo Stab_{A_{0}}. But this follows from Lemma 3.7, just as in the paragraph immediately following the proof of that lemma. □

Proof of Theorem 1.2. For the proof of Assertion (1), see the first paragraph of Section 1.3. The first inclusion in Assertion (2), namely B_{[A_{0}]}^{1,♯,2}(ε_{1}) ⊂ B_{[A_{0}]}^{1,∗,2}(c_{1}ε_{1}), follows from the definition (1.3) of the two distance functions defining the balls (1.4) and the Sobolev embedding L_{1}^{2} ⊂ L^{2♯,4} in Lemma 4.1. The second inclusion in Assertion (2), namely B_{[A_{0}]}^{1,∗,2}(c_{1}ε_{1}) ⊂ π(B_{A_{0}}^{4}(c_{2}K_{0}ε_{1})), follows from the definition (1.4) of these balls and Assertion (1) in Theorem 1.1. □

7. Critical-exponent Sobolev norms and the group of gauge transformations.

We now define an L_{2}^{♯,2} space of gauge transformations, by analogy with the definition of G_{E}^{k+1} when k ≥ 2, and set

G_{E}^{2,♯,2} := {u ∈ L_{2}^{♯,2}(𝔤𝔩(E)) : u ∈ G a.e.} ⊂ L_{k}^{2}(𝔤𝔩(E)).

It is not entirely clear a priori that G_{E}^{2,♯,2} is a Banach Lie group. In the case of its counterpart, G_{E}^{k+1}, the manifold structure follows from the fact that the exponential map

Exp : T_{id_{E}}G_{E} = Ω^{0}(𝔤_{E}) → G_{E}, ζ ↦ Exp ζ,

extends to a smooth map Exp : L_{k+1}^{2}(𝔤_{E}) → L_{k+1}^{2}(𝔤_{E}) and defines a system of smooth coordinate charts for G_{E}^{k+1}. Here, Exp is defined pointwise at u ∈ G_{E} for ζ ∈ T_{id_{E}}G_{E} by setting

(Exp_{u} ζ)(x) := exp_{u(x)}(ζ(x)), x ∈ X,

where exp : 𝔤 → G is the usual, C^{∞} exponential map for the Lie group G on the right-hand side [10, Appendix A].

To verify that G_{E}^{2,♯,2} is in fact a Banach Lie group we will need estimates for the covariant derivatives of the exponential map. The estimates below follow by reworking the usual proof of the Sobolev lemma for left composition of Sobolev sections by smooth vector bundle maps [15, Lemma 9.9]. The difference here is that we keep track of the dependence of the constants on the geometric data: This precision is required for the implicit function argument in the next section in order to complete the proof of our slice theorem.

For χ, ζ, ξ ∈ Ω^{0}(𝔤_{E}), the differentials

(D Exp)_{χ} : Ω^{0}(𝔤_{E}) → T_{Exp χ}G_{E}, ζ ↦ (D Exp)_{χ}ζ,
(D^{2} Exp)_{χ,ζ} : Ω^{0}(𝔤_{E}) → T_{Exp χ}G_{E}, ξ ↦ (D^{2} Exp)_{χ,ζ}ξ,

are defined pointwise by setting

(D Exp)_{χ}ζ|_{x} = (D exp)_{χ(x)}ζ(x),
(D^{2} Exp)_{χ,ζ}ξ|_{x} = (D exp)_{χ(x),ζ(x)}ξ(x),

for any x ∈ X. When writing the differential (D^{2} Exp)_{χ,ζ} above, we have identified T_{(D exp)_{χ}ζ}(T_{Exp χ}G_{E}) with T_{Exp χ}G_{E}.

The maps (D Exp)_{χ} : Ω^{0}(𝔤_{E}) → Ω^{0}(𝔤_{E}) and (D^{2} Exp)_{χ,ζ} : Ω^{0}(𝔤_{E}) → Ω^{0}(𝔤_{E}) extend linearly to maps

(D Exp)_{χ} : C^{∞}(⊗^{ℓ}(T^{∗}X) ⊗ 𝔤_{E}) → C^{∞}(⊗^{ℓ}(T^{∗}X) ⊗ 𝔤_{E}),
(D^{2} Exp)_{χ} : C^{∞}(⊗^{ℓ}(T^{∗}X) ⊗ 𝔤_{E}) → C^{∞}(⊗^{ℓ}(T^{∗}X) ⊗ 𝔤_{E}),

for ℓ ≥ 1, by setting

(D Exp)_{χ}(θ ⊗ ζ) = θ ⊗ (D Exp)_{χ}ζ,
(D^{2} Exp)_{χ,ζ}(θ ⊗ ξ) = θ ⊗ (D^{2} Exp)_{χ,ζ}ξ,

for θ ∈ ⊗^{ℓ}(T^{∗}X) and ξ ∈ Ω^{0}(𝔤_{E}). As usual, we embed G_{E} ⊂ Ω^{0}(𝔤_{E}) in order to compute the covariant derivatives of sections u ∈ G_{E}.

Lemma 7.1. Let G be a compact Lie group. Then there is a positive constant c(G) with the following significance. Let X be a closed, smooth, Riemannian four-manifold. If A is a C^{∞} connection on a G bundle E, and χ ∈ Ω^{0}(𝔤_{E}), then we have pointwise bounds:

(1)  |∇_{A}e^{χ}| ≤ |∇_{A}χ| + c|χ||∇_{A}χ|,
(2)  |∇_{A}^{2}e^{χ}| ≤ c(|χ| + |∇_{A}χ|)|∇_{A}χ| + c(1 + |χ|)|∇_{A}^{2}χ|,
(3)  |∇_{A}^{∗}∇_{A}e^{χ}| ≤ c(|χ| + |∇_{A}χ|)|∇_{A}χ| + c(1 + |χ|)|∇_{A}^{∗}∇_{A}χ|.

Proof. We have

∇_{A}e^{χ} = ∇_{A}(Exp χ) = (D Exp)_{χ} ∘ d_{A}χ ∈ Ω^{1}(𝔤_{E}).

Since (D exp)_{0} = id_{𝔼} and exp : 𝔤 → G is analytic, we have the pointwise bound |(D exp)_{χ(x)} − id_{𝔼}| ≤ c(G)|χ(x)| and thus a pointwise bound

|(D Exp)_{χ} − id_{E}| ≤ c|χ|,

noting that (D Exp)_{0} = id_{E}. Therefore, we have

|∇_{A}e^{χ}| ≤ |∇_{A}χ| + c|χ||∇_{A}χ|,

which gives the first assertion.

Define Φ(χ, ζ) = (D Exp)_{χ}(ζ) ∈ Ω^{1}(𝔤_{E}), for χ ∈ Ω^{0}(𝔤_{E}) and ζ ∈ Ω^{1}(𝔤_{E}), noting that Φ is nonlinear in χ, but linear in ζ. Thus,

∇_{A}^{2}u = (D_{1}Φ)_{(χ,∇_{A}χ)}(∇_{A}χ) + (D_{2}Φ)_{χ}(∇_{A}^{2}χ),

where D_{i}Φ, i = 1, 2, denote the partial derivatives of Φ with respect to first and second variables. Since (DΦ)_{(0,0)} = (D^{2} Exp)_{(0,0)} = id_{E}, as (D^{2} exp)_{0,0} = id_{𝔼}, and exp : 𝔤 → G is analytic we have the pointwise bound

|∇_{A}^{2}u| ≤ c(|χ| + |∇_{A}χ|)|∇_{A}χ| + c(1 + |χ|)|∇_{A}^{2}χ|,

giving the second assertion. Similarly, as ∗Φ(χ, ζ) = Φ(χ, ∗ζ) and ∇_{A}^{∗}∇_{A}u = − ∗ ∇_{A} ∗ ∇_{A}u, we have

|∇_{A}^{∗}∇_{A}u| ≤ c(|χ| + |∇_{A}χ|)|∇_{A}χ| + c(1 + |χ|)|∇_{A}^{∗}∇_{A}χ|,

giving the third assertion. □

The preceding pointwise bounds for ∇_{A}u, ∇_{A}^{2}u, and ∇_{A}^{∗}∇_{A}u yield the following estimates for the exponential map:

Lemma 7.2. Continue the hypotheses of Lemma 7.1. If k ≥ 2 is an integer (so L_{k+1}^{2} ⊂ C^{0}), A is an L_{k}^{2} connection on a G bundle E, and χ ∈ L_{k+1}^{2}(𝔤_{E}), then e^{χ} ∈ G_{E}^{k+1} satisfies

(1)  ∥∇_{A}e^{χ}∥_{L^{2}(X)} ≤ ∥∇_{A}χ∥_{L^{2}(X)} + c∥χ∥_{C^{0}(X)}∥∇_{A}χ∥_{L^{2}(X)},
(2)  ∥∇_{A}e^{χ}∥_{L^{♯}(X)} ≤ ∥∇_{A}χ∥_{L^{♯}(X)} + c∥χ∥_{C^{0}(X)}∥∇_{A}χ∥_{L^{♯}(X)},
(3)  ∥∇_{A}^{2}e^{χ}∥_{L^{2}(X)} ≤ c∥χ∥_{C^{0}(X)}∥∇_{A}χ∥_{L^{2}(X)} + ∥∇_{A}χ∥_{L^{4}(X)}^{2}
+ c(|1 + ∥χ∥_{C^{0}(X)})∥∇_{A}^{2}χ∥_{L^{2}(X)},
(4)  ∥∇_{A}^{∗}∇_{A}e^{χ}∥_{L^{♯}(X)} ≤ c∥χ∥_{C^{0}(X)}∥∇_{A}χ∥_{L^{♯}(X)} + ∥∇_{A}χ∥_{L^{2♯}(X)}^{2}
+ c(|1 + ∥χ∥_{C^{0}(X)})∥∇_{A}^{∗}∇_{A}χ∥_{L^{♯}(X)}.

The bounds (1)-(4) continue to hold for χ ∈ L_{2}^{♯,2}(𝔤_{E}) ⊂ C^{0}(𝔤_{E}), with A an L_{1}^{♯,2} connection on E, and Exp : Ω^{0}(𝔤_{E}) → G_{E} extends to a continuous map Exp : L_{2}^{♯,2}(𝔤_{E}) → G_{E}^{2,♯,2}.

Let A_{E}^{1,♯,2} = A_{0} + L_{1,A_{0}}^{♯,2}(Λ^{1} ⊗ 𝔤_{E}), for any C^{∞} reference connection A_{0} on E. Recall that we have an embedding L_{2}^{♯,2}(𝔤_{E}) ⊂ C^{0}(𝔤_{E}) and that the space L_{2}^{♯,2}(𝔤_{E}) is an algebra, while L_{1}^{♯,2}(Λ^{1} ⊗ 𝔤_{E}) and L_{1}^{2}(Λ^{1} ⊗ 𝔤_{E}) are L_{2}^{♯,2}(𝔤_{E})-modules. Therefore, the proofs of Propositions (A.2) and (A.3) in [10] extend to give the following analogue for G_{E}^{2,♯,2} in place of G_{E}^{k+1}:

Lemma 7.3. Let X be a closed Riemannian four-manifold and let E be a Hermitian vector bundle over X. Then the following hold.

(1) The space G_{E}^{2,♯,2} is a Banach Lie group with Lie algebra T_{id_{E}}G_{E}^{2,♯,2} = L_{2}^{♯,2}(𝔤_{E});

(2) The action of G_{E}^{2,♯,2} on A_{E}^{1} and A_{E}^{1,♯,2} is smooth;

(3) For A ∈ A_{E}^{1,♯,2}, the differential, at the identity id_{E} ∈ G_{E}^{2,♯,2}, of the map G_{E}^{2,♯,2} → A_{E}^{1,♯,2} given by u ↦ u(A) = A − (d_{A}u)u^{−1} is ζ ↦ −d_{A}ζ as a map L_{2}^{2,♯}(𝔤_{E}) → L_{1}^{♯,2}(Λ^{1} ⊗ 𝔤_{E}), and similarly for A ∈ A_{E}^{1}.

8. Existence of gauge transformations via the inverse function theorem.

Our goal in this section is to give an alternative, “direct” proof of Theorem 6.1 via the inverse function theorem. A direct argument—due to our overarching constraint that the constants given there ultimately depend at most on the L^{2} norm of the curvature and the least positive eigenvalue ν_{0}[A_{0}]—appears to be difficult within the standard framework of L_{2}^{p} (p > 2) gauge transformations acting on L_{1}^{p} connections; if this constraint is dropped then a direct proof is standard. However, we shall see that a direct argument is fairly straightforward within the framework of L_{2}^{♯,2} gauge transformations.

We already know that π(B_{A_{0}}^{4}(ε_{0})) is open in B_{E}^{k}, so it necessarily contains an L_{k,A_{0}}^{2}-ball centered at [A_{0}]. Via the inverse function theorem we estimate the radii of L_{1,A_{0}}^{♯,2} and L_{1,A_{0}}^{♯,2} balls, B_{[A_{0}]}^{1,∗,2}(ε) and B_{[A_{0}]}^{1,♯,2}(ε), which are contained in π(B_{A_{0}}^{4}(ε_{0})). Let us first dispose of the question of regularity for solutions to the second-order gauge-fixing equation:

Lemma 8.1. Let X be a closed, Riemannian four-manifold. Then there is a constant ε with the following significance. Let G be compact Lie group and let k ≥ 2 be an integer. Suppose that A_{0} is an L_{k}^{2} connection on a G bundle E, that a ∈ L_{k}^{2}(Λ^{1} ⊗ 𝔤_{E}) and χ ∈ L_{2}^{♯}(𝔤_{E}), and that u = e^{χ} is a solution to

d_{A_{0}}^{∗} ((d_{A_{0}}u)u^{−1} − uau^{−1}) = 0.

If ∥d_{A_{0}}u∥_{L^{4}} < ε then χ ∈ L_{k+1}^{2}(𝔤_{E}) and u = e^{χ} ∈ G_{E}^{k+1}.

Proof. Differentiation and right multiplication by u yields

d_{A_{0}}^{∗}d_{A_{0}}u + ∗((∗d_{A_{0}}u) ∧ u^{−1}d_{A_{0}}u) + ∗(d_{A_{0}}u ∧ ∗a) + ud_{A_{0}}^{∗}a (8.1)

+ ∗(ua ∧ ∗u^{−1}d_{A_{0}}u) = 0.

From Lemma 4.2 we know that u ∈ C^{0}∩L_{1}^{2} and so the last four terms in (8.1) are in L^{2}. Hence, d_{A_{0}}^{∗}d_{A_{0}}u is in L^{2} and so u ∈ L_{2}^{2} by elliptic regularity for d_{A_{0}}^{∗}d_{A_{0}}. The Sobolev embedding L_{1}^{2} ⊂ L^{4} and multiplication L^{4} × L^{q} → L^{p} for 2 ≤ p < 4 and 1/p = 1/4 + 1/q (so 4 ≤ q < ∞) now show that the last three terms in (8.1) are in L^{p}, so the equation takes the simpler form

d_{A_{0}}^{∗}d_{A_{0}}u + ∗((∗d_{A_{0}}u) ∧ u^{−1}d_{A_{0}}u) = v, (8.2)

where v ∈ L^{p}(𝔤_{E}) is the tautologically defined right-hand side and u ∈ L^{∞} ∩ L_{2}^{2}. Setting b = d_{A_{0}}u and noting that d_{A_{0}}b = F_{A_{0}}u, with F_{A_{0}} ∈ L_{k−1}^{2}(Λ^{2} ⊗ 𝔤_{E}) ⊂ L_{1}^{2}(Λ^{2} ⊗ 𝔤_{E}) and F_{A_{0}}u ∈ L_{1}^{2}(Λ^{2} ⊗ 𝔤_{E}). Thus, we may conveniently rewrite (8.2) as a first-order elliptic equation in b ∈ L_{1}^{2}(Λ^{1}⊗𝔤_{E}),

(d_{A_{0}}^{∗} + d_{A_{0}})b + ∗((∗b) ∧ u^{−1}b) = v′ ∈ L^{p}(𝔤_{E}) ⊕ L^{p}(Λ^{2} ⊗ 𝔤_{E}), (8.3)

where 2 < p < 4 and v′ = F_{A_{0}}u + v. Finally, (8.3) can be rewritten as a local equation by writing A_{0} = Γ + a_{0}, where Γ is the product connection in a local trivialization for E over a small ball U ⊂ X. Thus, the operator d_{A_{0}}^{∗} +d_{A_{0}} is replaced by d^{∗}+d in (8.2) and the additional terms are absorbed into the L^{p} inhomogeneous term v′ to give:

(d^{∗} + d)b + ∗((∗b) ∧ u^{−1}b) = v′′ ∈ L^{p}(U, 𝔤_{E}) ⊕ L^{p}(U, Λ^{2} ⊗ 𝔤_{E}). (8.4)

This is a first-order, elliptic equation with a quadratic non-linearity and Proposition 3.10 in [6] implies that the solution b = d_{A_{0}}u ∈ L_{1}^{2}(U, Λ^{1} ⊗ 𝔤_{E}) is necessarily in L_{1}^{p}(U′, Λ^{1} ⊗ 𝔤_{E}) for U′ ⋐ U, provided ∥b∥_{L^{4}(U)} < ε(g, p, U), and so u ∈ L_{2}^{p}(U′, 𝔤_{E}). In particular, we find that b ∈ L_{1}^{p}(X, Λ^{1} ⊗ 𝔤_{E}) and u ∈ L_{2}^{p}(X, 𝔤_{E}) for any 2 < p < 4, provided ∥d_{A_{0}}u∥_{L^{4}} < ε(g, p, X). The bootstrapping argument of Proposition 3.3 in [6] now implies that d_{A_{0}}u ∈ L_{k}^{2}(X, Λ^{1} ⊗ 𝔤_{E}). Thus u ∈ G_{E}^{k+1} and χ ∈ L_{k+1}^{2}(X, 𝔤_{E}), as desired. □

We can now proceed to the main argument:

Theorem 8.2. Let X be a closed, Riemannian four-manifold and let G be compact Lie group. Then there are positive constants c, z with the following significance. Let E be a G bundle over X and suppose that that A_{0} ∈ A_{E}^{2}, let K_{0}[A_{0}] = (1 + ν_{0}[A_{0}]^{−1})(1 + ∥F_{A_{0}}∥_{L^{2}}) and let ε_{1} be a constant satisfying

0 < ε_{1} ≤ zK_{0}^{−2}.

If A ∈ A_{E}^{2} obeys ∥A − A_{0}∥_{L_{1,A_{0}}^{♯,2}} < ε_{1} then u ∈ G_{E}^{3} exists such that

• d_{A_{0}}^{∗}(u(A) − A_{0}) = 0;

• ∥u(A) − A_{0}∥_{L_{1,A_{0}}^{2}} ≤ cK_{0}∥A − A_{0}∥_{L_{1,A_{0}}^{♯,2}} ;

• ∥u − id_{E}∥_{L_{2,A_{0}}^{♯,2}} < cK_{0}∥A − A_{0}∥_{L_{1,A_{0}}^{♯,2}} .

Proof. The argument is broadly similar to that of Lemma 3.6, except that we can show Ψ is a diffeomorphism directly—rather than just a local diffeomorphism—using the slightly stronger norms now at our disposal. Moreover, on this occasion we seek precise bounds on the solutions so we keep track of the dependence of constants on the curvature F_{A_{0}} and the least positive eigenvalue ν_{0} = ν_{0}[A_{0}] of the Laplacian Δ_{A_{0}} = d_{A_{0}}^{∗}d_{A_{0}}.

Write A = A_{0} + a and observe that

u(A) − A_{0} = A − A_{0} − (d_{A}u)u^{−1} = uau^{−1} − (d_{A_{0}}u)u^{−1}.

Recall that we have an L^{2}-orthogonal decomposition

Ω^{0}(𝔤_{E}) = (Ker d_{A_{0}})^{⊥} ⊕ Ker d_{A_{0}} = Im d_{A_{0}}^{∗} ⊕ Ker d_{A_{0}},

and that d_{A_{0}}^{∗} : L_{1}^{2}(Λ^{1} ⊗ 𝔤_{E}) → L^{2}(𝔤_{E}) has closed range; this gives

L_{2;A_{0}}^{♯,2}(𝔤_{E}) = (Ker d_{A_{0}}|_{L_{2;A_{0}}^{♯,2}} )^{⊥} ⊕ Ker d_{A_{0}}|_{L_{2;A_{0}}^{♯,2}}

= (Ker d_{A_{0}}^{∗}|_{L_{2;A_{0}}^{♯,2}} )^{⊥} ⊕ (Im d_{A_{0}}^{∗}|_{L_{1,A_{0}}^{♯,2}} ) .

We have a similar L^{2}-orthogonal decomposition

Ω^{1}(𝔤_{E}) = Im d_{A_{0}} ⊕ Ker d_{A_{0}}^{∗} = (Ker d_{A_{0}}^{∗})^{⊥} ⊕ Ker d_{A_{0}}^{∗},

and d_{A_{0}} : L_{1}^{2}(𝔤_{E}) → L^{2}(Λ^{1} ⊗ 𝔤_{E}) has closed range; this leads to the L^{2}- orthogonal decomposition

L_{1,A_{0}}^{♯,2}(Λ^{1} ⊗ 𝔤_{E}) = (Im d_{A_{0}}|_{L_{2;A_{0}}^{♯,2}} ) ⊕ (Ker d_{A_{0}}^{∗}|_{L_{1,A_{0}}^{♯,2}} )

= (Ker d_{A_{0}}^{∗}|_{L_{2;A_{0}}^{♯,2}} )^{⊥} ⊕ (Ker d_{A_{0}}^{∗}|_{L_{1,A_{0}}^{♯,2}} ) .

We now define a map

(8.5)  Ψ : (Ker (d_{A_{0}}|_{L_{2}^{♯,2}}))^{⊥} ⊕ Ker (d_{A_{0}}^{∗}|_{L_{1}^{♯,2}}) → L_{1,A_{0}}^{♯,2}(Λ^{1} ⊗ 𝔤_{E}),
(χ, a) ↦ uau^{−1} − (d_{A_{0}}u)u^{−1},

where u = e^{χ} and the differential at (χ, a) given by

(8.6) (DΨ)_{(χ,a)} : (Ker (d_{A_{0}}|_{L_{2}^{♯,2}}))^{⊥} ⊕ Ker (d_{A_{0}}^{∗}|_{L_{1}^{♯,2}}) → L_{1}^{♯,2}(Λ^{1} ⊗ 𝔤_{E}),
(ζ, b) ↦ u(−d_{A} ⊕ ι)u^{−1}(ζ, b) = u(−d_{A}ζ + b)u^{−1},

since (DΨ)_{(0,a)}(ζ, b) = −d_{A}ζ + b and Ψ is G_{E}-equivariant. Moreover, we have

(D^{2}Ψ)_{(χ,a)}((ζ, b), (η, α)) = u[η, −d_{A}ζ + b]u^{−1} + u[α, ζ]u^{−1}, (8.7)

for (ζ, b), (η, α) ∈ (Ker (d_{A_{0}}|_{L_{2}^{♯,2}}))^{⊥} ⊕ Ker (d_{A_{0}}^{∗}|_{L_{1}^{♯,2}}).

We now verify that the conditions of the inverse function theorem (Theorem 3.2) hold for suitable constants K and δ. The operator

d_{A_{0}} : (Ker (d_{A_{0}}|_{L_{2}^{♯,2}}))^{⊥} → (Ker (d_{A_{0}}^{∗}|_{L_{1}^{♯,2}}))^{⊥}

has a two-sided inverse

G_{A_{0}}^{0}d_{A_{0}}^{∗} : (Ker (d_{A_{0}}^{∗}|_{L_{1}^{♯,2}}))^{⊥} → (Ker (d_{A_{0}}|_{L_{2}^{♯,2}}))^{⊥} .

Indeed, for b ∈ (Ker (d_{A_{0}}^{∗}|_{L_{1}^{♯,2}}))^{⊥}, we have

∥G_{A_{0}}^{0}d_{A_{0}}^{∗}b∥_{L_{2,A_{0}}^{♯,2}} ≤ c_{0}K_{0}∥d_{A_{0}}^{∗}b∥_{L^{♯,2}} ≤ c_{0}K_{0}∥b∥_{L_{1,A_{0}}^{♯,2}} ,

and so G_{A_{0}}^{0}d_{A_{0}}^{∗} has Hom (L_{1,A_{0}}^{♯,2}, L_{2,A_{0}}^{♯,2}) operator norm bound

∥G_{A_{0}}^{0}d_{A_{0}}^{∗}∥ ≤ c_{0}K_{0}.

In particular, we see that (DΨ)_{(0,0)}^{−1} = G_{A_{0}}^{0}d_{A_{0}}^{∗} ⊕ id satisfies

∥(DΨ)_{(0,0)}^{−1}∥ ≤ c_{0}K_{0} (8.8)

the first of the conditions we need to verify for (DΨ)_{(0,0)} in order to apply the inverse function theorem.

It remains to compare (DΨ)_{(χ,a)} and (DΨ)_{(0,0)} using the mean value theorem,

(DΨ)_{(χ,a)}(ζ, b) − (DΨ)_{(0,0)}(ζ, b) = ∫_{0}^{1}(D^{2}Ψ)_{(tχ,ta)}((ζ, b), (χ, a)) dt. (8.9)

Thus, we need an estimate for D^{2}Ψ:

Claim 8.3. There is a universal polynomial function f(x, y), depending only on (X, g) and G, with f(0, 0) = 0, such that the following holds. For any t ∈ [0, 1] we have:

∥(D^{2}Ψ)_{(tχ,ta)}((ζ, b), (χ, a))∥_{L_{1,A_{0}}^{♯,2}}
≤ f (∥χ∥_{L_{2,A_{0}}^{♯,2}} , ∥a∥_{L_{1,A_{0}}^{♯,2}} ) (∥ζ∥_{L_{2,A_{0}}^{♯,2}} + ∥b∥_{L_{1,A_{0}}^{♯,2}} ) .

Proof. From (8.7) we have the L^{♯,2} estimate

∥(D^{2}Ψ)_{(tχ,ta)}((ζ, b), (χ, a))∥_{L^{♯,2}} ≤ c∥χ∥_{C^{0}} (∥d_{A_{0}}ζ∥_{L^{♯,2}} + ∥a∥_{L^{♯,2}}∥ζ∥_{C^{0}} + ∥b∥_{L^{♯,2}}) + c∥a∥_{L^{♯,2}}∥ζ∥_{C^{0}},

and thus:

∥(D^{2}Ψ)_{(tχ,ta)}((ζ, b), (χ, a))∥_{L^{♯,2}} (8.10)

≤ c (∥χ∥_{L_{2,A_{0}}^{♯,2}} + ∥a∥_{L^{♯,2}}∥χ∥_{L_{2,A_{0}}^{♯,2}} + ∥a∥_{L^{♯,2}}) (∥ζ∥_{L_{2,A_{0}}^{♯,2}} + ∥b∥_{L^{♯,2}}) .

The L^{2} estimate of ∇_{A_{0}}(D^{2}Ψ)_{(tχ,ta)}((ζ, b), (χ, a)) is given by

∥∇_{A_{0}}(D^{2}Ψ)_{(tχ,ta)}((ζ, b), (χ, a))∥_{L^{2}} ≤ c (∥∇_{A_{0}}u∥_{L^{4}}∥χ∥_{C^{0}} + ∥∇_{A_{0}}χ∥_{L^{4}}) (∥d_{A_{0}}ζ∥_{L^{4}} + ∥a∥_{L^{4}}∥ζ∥_{C^{0}} + ∥b∥_{L^{4}})

+ c∥χ∥_{C^{0}} (∥∇_{A_{0}}^{2}ζ∥_{L^{2}} + ∥∇_{A_{0}}a∥_{L^{2}}∥ζ∥_{C^{0}} + ∥a∥_{L^{4}}∥∇_{A_{0}}ζ∥_{L^{4}} + ∥∇_{A_{0}}b∥_{L^{2}}) + c∥∇_{A_{0}}u∥_{L^{4}}∥a∥_{L^{4}}∥ζ∥_{L^{4}} + c∥∇_{A_{0}}a∥_{L^{2}}∥ζ∥_{C^{0}} + c∥a∥_{L^{4}}∥∇_{A_{0}}ζ∥_{L^{4}},

and hence, using Lemma 7.2 to estimate u = e^{χ} in terms of χ,

∥∇_{A_{0}}(D^{2}Ψ)_{(tχ,ta)}((ζ, b), (χ, a))∥_{L^{2}} (8.11)

≤ f_{1} (∥χ∥_{L_{2,A_{0}}^{♯,2}} , ∥a∥_{L_{1,A_{0}}^{2}} ) (∥ζ∥_{L_{2,A_{0}}^{♯,2}} + ∥b∥_{L_{1,A_{0}}^{2}} ) ,

where f_{1}(x, y) is a polynomial function with f_{1}(0, 0) = 0.

Noting that d_{A_{0}}^{∗}a = 0, we have

(8.12)  d_{A_{0}}^{∗}[a, ζ] = d_{A_{0}}^{∗}(aζ − ζa)
= (d_{A_{0}}^{∗}a)ζ − a ∧ d_{A_{0}}ζ − ∗(d_{A_{0}}ζ ∧ ∗a) − ζ(d_{A_{0}}^{∗}a)
= −a ∧ d_{A_{0}}ζ − ∗(d_{A_{0}}ζ ∧ ∗a),

and similarly for d_{A_{0}}^{∗}[χ, b] since d_{A_{0}}^{∗}b = 0. For any β ∈ L_{1}^{2}(Λ^{1} ⊗ 𝔤_{E}) we have

(8.13)  d_{A_{0}}^{∗}(uβu^{−1}) = − ∗ d_{A_{0}}(u(∗β)u^{−1}))
= − ∗ (d_{A_{0}}u ∧ ∗βu^{−1}) + u(d_{A_{0}}^{∗}β)u^{−1}
− ∗u((∗β) ∧ u(d_{A_{0}}u)u^{−1}).

Therefore, Equations (8.7), (8.12), and (8.13) and the estimates for u = e^{χ} in Lemma 7.2 yield

∥d_{A_{0}}^{∗}(D^{2}Ψ)_{(χ,a)}((ζ, b), (χ, a))∥_{L^{♯,2}} (8.14)

≤ ∥d_{A_{0}}^{∗} (u[χ, −d_{A}ζ + b]u^{−1} + u[a, ζ]u^{−1})∥_{L^{♯,2}}
≤ f_{2} (∥χ∥_{L_{2,A_{0}}^{♯,2}} , ∥a∥_{L^{2♯,4}}) (∥ζ∥_{L_{2,A_{0}}^{♯,2}} + ∥b∥_{L^{2♯,4}}) ,

where f_{2}(x, y) is a polynomial function with f_{2}(0, 0) = 0. The claim now follows by combining (8.10), (8.11), and (8.14). □

Therefore, from Claim 8.3 and (8.9) we have

∥(DΨ)_{(χ,a)}(ζ, b) − (DΨ)_{(0,0)}(ζ, b)∥_{L_{1,A_{0}}^{♯,2}} (8.15)

≤ f (∥χ∥_{L_{2,A_{0}}^{♯,2}} , ∥a∥_{L_{1,A_{0}}^{♯,2}} ) (∥ζ∥_{L_{2,A_{0}}^{♯,2}} + ∥b∥_{L_{1,A_{0}}^{♯,2}} ) .

Consequently, with respect to the Hom (L_{2,A_{0}}^{♯,2}, L_{1,A_{0}}^{♯,2}) operator norm, (8.15) yields the bound

∥(DΨ)_{(χ,a)} − (DΨ)_{(0,0)}∥ ≤ \frac{1}{2}c_{0}^{−1}K_{0}^{−1}, (8.16)

where c_{0}K_{0} = K is the constant of (8.8), provided (χ, a) satisfies the constraint

∥χ∥_{L_{2,A_{0}}^{♯,2}} + ∥a∥_{L_{1,A_{0}}^{♯,2}} ≤ c_{1}K_{0}^{−1} = δ. (8.17)

Define balls centered at the origins in (Ker (d_{A_{0}}|_{L_{2}^{♯,2}}))^{⊥} and Ker (d_{A_{0}}^{∗}|_{L_{1}^{♯,2}}) by setting

B_{0}^{⊥;2,♯,2}(δ) = {χ ∈ (Ker (d_{A_{0}}|_{L_{2}^{♯,2}}))^{⊥} : ∥χ∥_{L_{2,A_{0}}^{♯,2}} < δ} ,
B_{0}^{1,♯,2}(δ) = {a ∈ Ker (d_{A_{0}}^{∗}|_{L_{1}^{♯,2}}) : ∥a∥_{L_{1,A_{0}}^{♯,2}} < δ} .

Hence, Theorem 3.2 implies that the map

Ψ : B_{0}^{⊥;2,♯,2}(δ) × B_{0}^{1,♯,2}(δ) → A_{E}^{1,♯,2}

is injective, its image is an open subset of A_{E}^{1,♯,2} and contains the ball B_{A_{0}}^{1,♯,2}(δ/(2K)), the inverse map Ψ^{−1} is a diffeomorphism from B_{A_{0}}^{1,♯,2}(δ/(2K)) onto its image, and if (χ_{1}, A_{1}), (χ_{2}, A_{2}) are points in B_{0}^{⊥,2,♯,2}(δ) × B_{0}^{1,♯,2}(δ), then

∥χ_{1} − χ_{2}∥_{L_{2,A_{0}}^{♯,2}} + ∥A_{1} − A_{2}∥_{L_{1,A_{0}}^{♯,2}} ≤ 2K∥u_{1}(A_{1}) − u_{2}(A_{2})∥_{L_{1,A_{0}}^{♯,2}} ,

where u_{i} = e^{χ_{i}}, i = 1, 2. In particular, setting (χ_{2}, A_{2} − A_{0}) = (0, 0), we see that if A is a point in A_{E}^{1,♯,2} such that ∥A − A_{0}∥_{L_{1,A_{0}}^{♯,2}} < δ/(2K), then there

is a unique solution (χ, u^{−1}(A)) = Ψ^{−1}(A) in B_{0}^{⊥,2,♯,2}(δ) × B_{0}^{1,♯,2}(δ). Here, u = e^{χ} is a gauge transformation with χ ∈ B_{0}^{⊥;2,♯,2}(δ) such that

\begin{aligned}d_{A_{0}}^{∗}(u^{−1}(A) − A_{0}) & = 0, \\ ∥χ∥_{L_{2,A_{0}}^{♯,2}} + ∥u^{−1}(A) − A_{0}∥_{L_{1,A_{0}}^{♯,2}} & ≤ 2K∥A − A_{0}∥L_{1,A_{0}}^{♯,2}. \end{aligned} (8.18)

Lemma 7.2 implies that u = e^{χ} satisfies

∥u − id_{E}∥_{L_{2,A_{0}}^{♯,2}} ≤ f_{3} (∥χ∥_{L_{2,A_{0}}^{♯,2}} ) ≤ c∥χ∥_{L_{2,A_{0}}^{♯,2}} ≤ c_{2}δ, (8.19)

where f_{3}(x) is a polynomial with coefficients depending only on (X, g) and G such that f_{3}(0) = 0. Noting that K = c_{0}K_{0}, δ = c_{1}K_{0}^{−1}, and δ/(2K) = \frac{1}{2}c_{0}c_{1}K_{0}^{−2}, the desired estimates follows from (8.18) and (8.19). Finally, Lemma 8.1 implies that u ∈ G_{E}^{3} and this completes the proof of the theorem.

□

While the L_{1}^{2} estimate of Theorem 8.2 suffices for most purposes, it is occasionally useful to have the weaker L^{2♯,4} bound at hand. Recall from Section 4 that we defined

∥a∥_{L_{1,A_{0}}^{♯,2}} = ∥a∥_{L^{2♯,4}} + ∥d_{A_{0}}^{∗}a∥_{L^{♯,2}}, a ∈ Ω^{1}(𝔤_{E}).

A slight modification of the proof of Theorem 8.2 yields:

Theorem 8.4. Continue the hypotheses of Theorem 8.2. Then for any A ∈ A_{E}^{2} such that ∥A − A_{0}∥_{L_{1,A_{0}}^{♯,2}} < ε_{1} there is a gauge transformation u ∈ G_{E}^{3}

with the following properties:

• d_{A_{0}}^{∗}(u(A) − A_{0}) = 0;

• ∥u(A) − A_{0}∥_{L^{2♯,4}} ≤ cK_{0}∥A − A_{0}∥_{L_{1,A_{0}}^{♯,2}} ;

• ∥u − id_{E}∥_{L_{2,A_{0}}^{♯,2}} < cK_{0}∥A − A_{0}∥_{L_{1,A_{0}}^{♯,2}} .

Proof. The first difference in the argument is that the map Ψ in (8.5) is replaced by

Ψ : (Ker (d_{A_{0}}|_{L_{2}^{♯,2}}))^{⊥} ⊕ Ker (d_{A_{0}}^{∗}|_{L_{1}^{♯,2}}) → L_{1,A_{0}}^{♯,2}(Λ^{1} ⊗ 𝔤_{E}),
(χ, a) ↦ uau^{−1} − (d_{A_{0}}u)u^{−1}.

(8.20)

The second difference is that Claim 8.3 is replaced by:

Claim 8.5. There is a universal polynomial function f(x, y), depending only on (X, g) and G, with f(0, 0) = 0, such that the following holds. For any t ∈ [0, 1] we have:

∥(D^{2}Ψ)_{(tχ,ta)}((ζ, b), (χ, a))∥_{L_{1,A_{0}}^{♯,2}}
≤ f (∥χ∥_{L_{2,A_{0}}^{♯,2}} , ∥a∥_{L^{2♯,4}}) (∥ζ∥_{L_{2,A_{0}}^{♯,2}} + ∥b∥_{L^{2♯,4}}) .

Proof. From (8.7) we now have the L^{2♯,4} estimate

∥(D^{2}Ψ)_{(tχ,ta)}((ζ, b), (χ, a))∥_{L^{2♯,4}} ≤ c∥χ∥_{C^{0}} (∥d_{A_{0}}ζ∥_{L^{2♯,4}} + ∥a∥_{L^{2♯,4}}∥ζ∥_{C^{0}} + ∥b∥_{L^{2♯,4}}) + c∥a∥_{L^{2♯,4}}∥ζ∥_{C^{0}},

and thus:

∥(D^{2}Ψ)_{(tχ,ta)}((ζ, b), (χ, a))∥_{L^{2♯,4}} (8.21)

≤ c (∥χ∥_{L_{2,A_{0}}^{♯,2}} + ∥a∥_{L^{2♯,4}}∥χ∥_{L_{2,A_{0}}^{♯,2}} + ∥a∥_{L^{2♯,4}})
× (∥ζ∥_{L_{2,A_{0}}^{♯,2}} + ∥b∥_{L^{2♯,4}}) .

Combining (8.14) and (8.21) yields the claim. □

The rest of the argument is just as before. This completes the proof of the theorem. □

We now have our second proof of Theorem 6.1 via Theorems 8.2 and 8.4:

Proof of Theorem 6.1. From the hypotheses we have A_{0} ∈ A_{E}^{k} and [A] ∈ B_{E}^{k} with k ≥ 2. According to Lemma 6.3, there is gauge transformation w ∈ G_{E}^{3} such that

dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) = ∥w(A) − A_{0}∥_{L_{1,A_{0}}^{♯,2}} ,

where A ∈ A_{E}^{k}, so Theorems 8.2 and the argument of 8.4 imply that there is a gauge transformation v ∈ G_{E}^{3} so that u(A) satisfies the conclusions of Assertion (2) with u = vw ∈ G_{E}^{3}. Since d_{A_{0}}^{∗}(u(A) − A_{0}) = 0 and u ∈ G_{E}^{3} and A, A_{0} ∈ A_{E}^{k}, a standard bootstrapping argument implies that u ∈ G_{E}^{k+1}.

Similarly, by Lemma 6.3, there is gauge transformation w ∈ G_{E}^{3} such that

dist_{L_{1,A_{0}}^{♯,2}} ([A], [A_{0}]) = ∥w(A) − A_{0}∥_{L_{1,A_{0}}^{♯,2}} ,

so Assertion (1) follows from Theorem 8.4 in the same manner. □

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Received July 29, 1999 and revised August 10, 2000. The author was supported in part by an NSF Mathematical Sciences Postdoctoral Fellowship under grant DMS 9306061 and by NSF grant DMS 9704174.

Rutgers University Piscataway, NJ 08854-8019, USA E-mail address: feehan@math.rutgers.edu Current address: University of Dublin Trinity College Dublin 2, Ireland E-mail address: feehan@maths.tcd.ie

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