Convexity of the Potential Function of the Einstein-K\"ahler Metric on a Convex Domain

TL;DR

This paper proves that the potential function of a complete K"ahler-Einstein metric on a bounded strictly convex domain in complex space is itself strictly convex, revealing a fundamental geometric property.

Contribution

It establishes the strict convexity of the potential function for K"ahler-Einstein metrics on convex domains, a novel geometric insight.

Findings

Potential function u is strictly convex.

Supports geometric understanding of K"ahler-Einstein metrics.

Advances convexity theory in complex differential geometry.

Abstract

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