Strict convexity of the Mabuchi functional for energy minimizers

TL;DR

This paper proves the strict convexity of the Mabuchi functional along certain geodesics, using a new formula for the complex Hessian of the weighted log-Bergman kernel, with implications for energy minimizers.

Contribution

It introduces an explicit formula for the complex Hessian of the weighted log-Bergman kernel and applies it to establish strict convexity of the Mabuchi functional along specific geodesics.

Findings

Explicit formula for the complex Hessian of the weighted log-Bergman kernel.

Proof of strict convexity of the Mabuchi functional along smooth geodesics.

Strict convexity when connecting non-degenerate energy minimizers via C^{1,1}-geodesics.

Abstract

There are two parts of this paper. First, we discovered an explicit formula for the complex Hessian of the weighted log-Bergman kernel on a parallelogram domain, and utilised this formula to give a new proof about the strict convexity of the Mabuchi functional along a smooth geodesic. Second, when a C^{1,1}-geodesic connects two non-degenerate energy minimizers, we also proved this strict convexity, by showing that such a geodesic must be non-degenerate and smooth.