Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convexity of the Potential Function of the Einstein-Kähler Metric on a Convex Domain

Jingchen Hu, Li Sheng

Abstract

Suppose that u is the potential function of a complete Kähler-Einstein metric on a bounded strictly convex domain in $\mathbb{C}^n$. We prove that u itself is strictly convex.

Convexity of the Potential Function of the Einstein-Kähler Metric on a Convex Domain

Abstract

Suppose that u is the potential function of a complete Kähler-Einstein metric on a bounded strictly convex domain in . We prove that u itself is strictly convex.
Paper Structure (6 sections, 3 theorems, 39 equations)

This paper contains 6 sections, 3 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Proof of the Main Theorem
  3. Equations for Second Derivatives of $u$: $A, B$
  4. Computing $u^{{i\overline j}}{\partial_{{i\overline j}}} M$
  5. Proving the Result Using the Maximum Principle
  6. Linear Algebra Lemma

Key Result

Theorem 1

Suppose that $\Omega$ is a $C^k$, $k\geq \max(3n+6,2n+9)$, strictly pseudoconvex domain in $\mathbb{C}^n$. Then there is a function $v\in C^{\omega}(\Omega)\cap C^{n+3/2-\delta}(\overline{\Omega})$, for any $\delta>0$, solving (eq:potential_KE_exp).

Theorems & Definitions (4)

  • Theorem : Corollary 6.6 of ChengYau_80_KahlerMetric
  • Theorem 1.1
  • Lemma A.1
  • proof