Subharmonicity of conic Mabuchi's functional, I

TL;DR

This paper extends the convexity properties of Mabuchi's functional to conic Kähler metrics, establishing a framework for analyzing conic cscK metrics and proving the convexity of the conic Mabuchi functional along geodesics.

Contribution

It introduces a new framework for conic cscK metrics and proves the convexity of the conic Mabuchi functional along geodesics, generalizing previous results to the conic setting.

Findings

Established a framework for conic cscK metrics.

Introduced the conic Mabuchi functional.

Proved convexity of the conic Mabuchi functional along geodesics.

Abstract

The purpose of this paper is to generalize the convexity of Mabuchi's functional to the conic setting. We first established a frame to study conic cscK metrics, and then the conic Mabuchi functional was introduced in such a way that conic cscK metrics are its critical points. Finally we proved the convexity of the conic Mabuchi functional along any conic geodesics.