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Bergman representative coordinate, constant holomorphic curvature and a multidimensional generalization of Carathéodory's theorem

Robert Xin Dong, Bun Wong

TL;DR

This paper extends Lu's Lu Qi-Keng-type uniformization from complete Bergman metrics to Bergman-incomplete bounded pseudoconvex domains with constant negative curvature $-c^2$, using Bergman representative coordinates and Calabi's diastasis. It proves that such domains are biholomorphic to a ball $ olinebreak[4] olinebreak[4] olinebreak[4] olinebreak[4] olinebreak[4] igl o olinebreak[4] egin{Bmatrix}w: egin{aligned} extstyle rac{}{} rac{}{} rac{}{} rac{}{} rac{}{} rac{}{} rac{}{} ag{placeholder} rac{}{} rac{}{} rac{}{} rac{}{} rac{}{} rac{}{} rac{}{} rac{}{} rac{}{} egin{cases} ext{} ext{} ext{} ext{} ext{} ext{} ext{} ext{} ext{} ext{} ext{} ext{} ext{} ext{} ext{} ext{} ext{} ext{} ext{} ext{} ext{} ext{} ext{} ext{} ext{} ext{} ext{} ext{} ext{} ext{} ext{} ext{} egin{aligned} \ ag{ball} rac{2}{c^2} o d e a d d e d d d d d d d eq } e d d d d d d d d d d d d d d d d d d d d d d them, possibly minus a pluripolar set $E$, and that the associated biholomorphism $T$ extends to a homeomorphism of the closures, yielding that the boundary $ abla \partial ext{Ω}$ is homeomorphic to the sphere $S^{2n-1}$ under local $C^1$-connectivity. A Harnack-type estimate for the Bergman kernel is established, and a multidimensional Carathéodory-type extension is derived. The results unify Lu's and Suita's line of work in higher dimensions and connect to boundary regularity phenomena for $L^2$-domains of holomorphy. Applications include criteria ensuring boundary spherical topology for domains biholomorphic to balls and corollaries relating to boundary extension of Riemann maps in the planar case.

Abstract

By using the Bergman representative coordinate and Calabi's diastasis, we extend a theorem of Lu to bounded pseudoconvex domains whose Bergman metric is incomplete with constant holomorphic sectional curvature. We characterize such domains that are biholomorphic to a ball possibly less a relatively closed pluripolar set. We also provide a multidimensional generalization of Carathéodory's theorem on the continuous extension of the biholomorphisms up to the closures. In particular, sufficient conditions are given, in terms of the Bergman kernel, for the boundary of a biholomorphic ball to be a topological sphere.

Bergman representative coordinate, constant holomorphic curvature and a multidimensional generalization of Carathéodory's theorem

TL;DR

This paper extends Lu's Lu Qi-Keng-type uniformization from complete Bergman metrics to Bergman-incomplete bounded pseudoconvex domains with constant negative curvature , using Bergman representative coordinates and Calabi's diastasis. It proves that such domains are biholomorphic to a ball ET abla \partial ext{Ω}S^{2n-1}C^1L^2$-domains of holomorphy. Applications include criteria ensuring boundary spherical topology for domains biholomorphic to balls and corollaries relating to boundary extension of Riemann maps in the planar case.

Abstract

By using the Bergman representative coordinate and Calabi's diastasis, we extend a theorem of Lu to bounded pseudoconvex domains whose Bergman metric is incomplete with constant holomorphic sectional curvature. We characterize such domains that are biholomorphic to a ball possibly less a relatively closed pluripolar set. We also provide a multidimensional generalization of Carathéodory's theorem on the continuous extension of the biholomorphisms up to the closures. In particular, sufficient conditions are given, in terms of the Bergman kernel, for the boundary of a biholomorphic ball to be a topological sphere.
Paper Structure (8 sections, 17 theorems, 77 equations, 1 figure)

This paper contains 8 sections, 17 theorems, 77 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Remark.
  3. Proofs of Multidimensional Results
  4. Estimates of Bergman's Coordinates
  5. Proof of the Main Theorem
  6. $L^2$-Domain of Holomorphy
  7. Proofs of One Dimensional Results
  8. Bounded Bergman Representative Coordinates

Key Result

Theorem 1.2

Let $\Omega \subset \mathbb C$ be a bounded domain whose Bergman metric has Gaussian curvature identically equal to $- 2$. If there exists some point $p \in \Omega$ such that $\left| K(z, p) \right|$ is bounded from above by a finite constant $\mathcal{C}_1>0$ for any $z \in \Omega$, th

Figures (1)

  • Figure 1: A simply-connected domain with discontinuous boundary

Theorems & Definitions (35)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Proposition 2.1
  • proof
  • ...and 25 more