The Complex Analysis and Geometry Seminar will run on Fridays, 10:30-11:30 am, in Hill 705. All are welcome!

The seminar is organized by Xiaojun Huang and Purvi Gupta.

### Schedule of talks for Fall 2019

• September 27, 2019.
Debraj Chakrabarti, Central Michigan University

Title. TBA

Abstract. TBA

• September 27, 2019.
Note: This is a departmental colloquium talk.

Steven Krantz, Washington University, St. Louis
Time: 4:00 pm
Venue: HLL 705

Title.TBA

Abstract. TBA

### Past Talks

• October 05, 2018.
Sean Curry, University of California, San Diego

Title. Strictly pseudoconvex domains in $\mathbb{C}^2$ with obstruction flat boundary.

Abstract. A bounded strictly pseudoconvex domain in $\mathbb{C}^n$, $n>1$, supports a unique complete Kahler-Einstein metric determined by the Cheng-Yau solution of Fefferman's Monge-Ampere equation. The smoothness of the solution of Fefferman's equation up to the boundary is obstructed by a local curvature invariant of the boundary called the obstruction density. In the case $n=2$ the obstruction density is especially important, in particular in describing the logarithmic singularity of the Bergman kernel. For domains in $\mathbb{C}^2$ diffeomorphic to the ball, we motivate and consider the problem of determining whether the global vanishing of this obstruction implies biholomorphic equivalence to the unit ball. (This is a strong form of the Ramadanov Conjecture.)

• October 12, 2018.
Rasul Shafikov, University of Western Ontario, Canada

Title. On rationally convex embeddings and immersions of real manifolds in complex spaces.

Abstract. A classical result of Duval-Sibony characterizes rationally convex totally real embeddings of real manifolds into $\mathbb{C}^n$ as those that are isotropic with respect to some Kahler form. In this talk I will describe some generalizations of this result for topological embeddings and immersions, and will discuss some applications.

• February 15, 2019.
Shiferaw Berhanu, Temple University

Title. Unique continuation at the boundary for a class of elliptic equations

Abstract. We will present some recent results on unique continuation at the boundary for a class of second order elliptic partial differential equations and the biharmonic operator. The works are inspired by a theorem of X. Huang et al for holomorphic functions which also motivated M. S. Baouendi and L. Rothschild to prove generalizations for harmonic functions.

• March 08, 2019.
Xin Dong, University of California, Irvine

Title. Bergman and Suita metrics

Abstract. For any open Riemann surface $X$ admitting Green functions, the Suita conjecture states that the Gaussian curvature of the Suita metric induced by the logarithmic capacity is bounded from above by $-4$, and the curvature is equal to $-4$ at some point if and only if $X$ is biholomorphic to the unit disc less a (possibly) closed polar subset. We talk about our new proof of the above equality part by using the plurisubharmonic variation properties of the Bergman kernels. We also relate this with a joint work with Bun Wong on the holomorphic sectional curvature of the Bergman metric on manifolds.

• March 15, 2019.
Andrew Zimmer, Louisiana State University, Baton Rouge

Title. The geometry of domains with negatively pinched Kaehler metrics

Abstract. Every bounded pseudoconvex domain in $\mathbb{C}^n$ has a natural complete metric: the Kaehler-Einstein metric constructed by Cheng-Yau. When the boundary of the domain is strongly pseudoconvex, Cheng-Yau showed that the holomorphic sectional curvature of this metric is asymptotically a negative constant. In this talk I will describe some converses to this result, including the following: if a smoothly bounded convex domain has a complete Kaehler metric with close to constant negative holomorphic sectional curvature near the boundary, then the domain is strongly pseudoconvex. This is joint work with F. Bracci and H. Gaussier.

• April 12, 2019.
Ved Datar, Indian Institute of Science, Bangalore

Title. Adiabatic limits of ASD connections on collapsing K3 surfaces

Abstract.I will talk about recent joint work with Adam Jacob and Yuguang Zhang on a convergence result for a family of Yang-Mills connections over an elliptic K3 surface as the fibres collapse. We show that outside a finite number of fibres, the curvatures of the connections are uniformly bounded (with respect to fixed background metrics), and that the connections converge in L^{1,p} to a limiting connection whose restriction to a generic fibre is a flat connection. This solves a conjecture of Fukaya in dimension two, and is a vector bundle analog of the work of Gross-Wilson and Gross-Tosatti-Zhang on the collapsing behaviour of Ricci flat metrics in degenerating Kahler classes.

• April 19, 2019.
Berit Stensones, Norwegian University of Science and Technology

Title. Examples of Automorphisms with Fatou components with complicated topology

Abstract. It is known that in 1 variable the Fatou Component of an entire map will have to be simply connected. Further one can not have points that under iterations of a map spirals in toward a fixed point, the so called snail lemma. We shall show that in higher dimensions the situation is very different.

• April 26, 2019.
Note: Two talks on the same day.
Ben Weinkove, Northwestern University

Time: 10:30 am
Venue: HLL 705

Title. Complex Monge-Ampere equations on manifolds

Abstract. More than 40 years ago, Yau solved the complex Monge-Ampere equation on compact Kahler manifolds. I will give an overview of some extensions of this result including the case of non-Kahler manifolds and behavior of solutions when the equation degenerates. I will also indicate some open problems and new directions.

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John Erik Fornaess, Norwegian University of Science and Technology
Time: 2:30 pm
Venue: HLL 705

Title.The squeezing function

Abstract. In complex analysis the most important domain is the unit disc. In fact all domains (at least simply connected and bounded) are biholomorphic, i.e. analytically equivalent, to the disc. In higher dimension, the natural analogue is the unit ball. But in higher dimension, the general domain is not biholomorphic to the ball. A basic question is then how well a general domain can be approximated by the ball. If we have a ball B_r of radius r<1 contained in the unit ball B_1, then a domain U in B_1 containing B_r is said to be squeezed between the two balls. The larger we can choose r, the closer the domain U is to the ball.