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Kähler-Ricci solitons on bounded pseudoconvex domains

Zehao Sha

TL;DR

The paper establishes a rigidity phenomenon for Kähler-Ricci solitons on bounded pseudoconvex domains by showing that, under a $C^1$-bounded geometry assumption and a negatively pinched Ricci curvature outside a compact set, any soliton with a closed dual 1-form must be Kähler-Einstein. It strengthens the Bergman metric side of Cheng’s conjecture by leveraging Huang-Xiao’s results, proving that a Bergman Kähler-Ricci soliton forces the domain to be biholomorphic to the ball. The authors also develop a framework for uniformly squeezing domains and verify the KE rigidity in explicit model classes, notably Thullen and Hartogs domains, where the soliton condition collapses to KE and yields strong biholomorphic classification. Overall, the work provides a conditional rigidity theory for noncompact bounded domains, linking soliton structure, boundary geometry, and canonical metrics with well-known model geometries in several complex variables.

Abstract

In this paper, we study the Kähler-Ricci soliton on bounded pseudoconvex domains in $\mathbb{C}^n$. We proved that under certain assumptions, this soliton is a Kähler-Einstein metric. Building on the work of Huang and Xiao, we investigate an analogue of Cheng's conjecture for the Bergman Kähler-Ricci soliton. Additionally, we introduce some model domains as examples.

Kähler-Ricci solitons on bounded pseudoconvex domains

TL;DR

The paper establishes a rigidity phenomenon for Kähler-Ricci solitons on bounded pseudoconvex domains by showing that, under a -bounded geometry assumption and a negatively pinched Ricci curvature outside a compact set, any soliton with a closed dual 1-form must be Kähler-Einstein. It strengthens the Bergman metric side of Cheng’s conjecture by leveraging Huang-Xiao’s results, proving that a Bergman Kähler-Ricci soliton forces the domain to be biholomorphic to the ball. The authors also develop a framework for uniformly squeezing domains and verify the KE rigidity in explicit model classes, notably Thullen and Hartogs domains, where the soliton condition collapses to KE and yields strong biholomorphic classification. Overall, the work provides a conditional rigidity theory for noncompact bounded domains, linking soliton structure, boundary geometry, and canonical metrics with well-known model geometries in several complex variables.

Abstract

In this paper, we study the Kähler-Ricci soliton on bounded pseudoconvex domains in . We proved that under certain assumptions, this soliton is a Kähler-Einstein metric. Building on the work of Huang and Xiao, we investigate an analogue of Cheng's conjecture for the Bergman Kähler-Ricci soliton. Additionally, we introduce some model domains as examples.

Paper Structure

This paper contains 11 sections, 18 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Kähler-Einstein metric and Kähler-Ricci soliton
  5. Bounded geometry and Omori-Yau's maximum principle
  6. Proof of Theorem \ref{['MT1']}
  7. Strictly pseudocnvex domains
  8. Uniformly squeezing domains
  9. Examples
  10. Thullen domain in $\mathbb{C}^2$
  11. Hartogs domains

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{C}^n$ be a bounded pseudoconvex domain, and let $g$ be a complete Kähler metric on $\Omega$ of $C^1$-bounded geometry. Suppose there exists a compact subset $K \subset \subset \Omega$ such that $0> -C_1 g \geq \operatorname{Ric}(g)\geq -C_2 g$ in $\Omega \setminus K$ for

Theorems & Definitions (32)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Proposition 2.2: omori1967isometricyau1975harmonic
  • Remark 2.3
  • Proposition 3.1
  • proof
  • Lemma 3.2: alias2016maximum, Proposition 8.3
  • Proposition 3.3
  • proof
  • ...and 22 more