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Bergman-Einstein metric on a Stein space with a strongly pseudoconvex boundary

Xiaojun Huang, Xiaoshan Li

TL;DR

The paper generalizes Cheng–Yau–type rigidity for Bergman metrics to Stein spaces with compact, smooth, strongly pseudoconvex boundaries by proving: if the Bergman metric on the regular part is Kähler–Einstein, then the boundary is spherical. The approach combines a resolution of singularities, localization of Bergman kernel forms via Fefferman-type arguments, and Monge–Ampère equations for associated potentials, linking the KE condition to boundary CR geometry. It then analyzes ball quotients, showing nontrivial finite quotients typically fail to be KE, while in dimension one all such quotients are KE; this yields a Cheng-type rigidity result for Stein spaces with spherical boundary and clarifies the role of ball quotients in this context. Altogether, the work extends Cheng conjecture-type phenomena to singular settings and provides a criterion to detect ball domains from Bergman–Kähler–Einstein geometry.

Abstract

Let $Ω$ be a Stein space with a compact smooth strongly pseudoconvex boundary. We prove that the boundary is spherical if its Bergman metric over $\hbox{Reg}(Ω)$ is Kähler-Einstein.

Bergman-Einstein metric on a Stein space with a strongly pseudoconvex boundary

TL;DR

The paper generalizes Cheng–Yau–type rigidity for Bergman metrics to Stein spaces with compact, smooth, strongly pseudoconvex boundaries by proving: if the Bergman metric on the regular part is Kähler–Einstein, then the boundary is spherical. The approach combines a resolution of singularities, localization of Bergman kernel forms via Fefferman-type arguments, and Monge–Ampère equations for associated potentials, linking the KE condition to boundary CR geometry. It then analyzes ball quotients, showing nontrivial finite quotients typically fail to be KE, while in dimension one all such quotients are KE; this yields a Cheng-type rigidity result for Stein spaces with spherical boundary and clarifies the role of ball quotients in this context. Altogether, the work extends Cheng conjecture-type phenomena to singular settings and provides a criterion to detect ball domains from Bergman–Kähler–Einstein geometry.

Abstract

Let be a Stein space with a compact smooth strongly pseudoconvex boundary. We prove that the boundary is spherical if its Bergman metric over is Kähler-Einstein.

Paper Structure

This paper contains 5 sections, 14 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['mthm-4-18']}
  3. Localization of Bergman kernel forms
  4. Proof of Theorem \ref{['mthm1']}
  5. Bergman metric on a ball quotient

Key Result

Theorem 1.1

Let $\Omega$ be a Stein space with a compact smooth strongly pseudoconvex boundary. If its Bergman metric $\omega_\Omega^B$ on ${\rm Reg}(\Omega)$ is Kähler-Einstein then $\partial\Omega$ is spherical.

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 2.1
  • Corollary 2.2
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • ...and 15 more