Approximation of weak geodesics and subharmonicity of Mabuchi energy

TL;DR

This paper investigates the convexity and continuity of the Mabuchi functional along weak geodesics, using a novel approximation method involving Monge-Ampère equations to analyze geometric energy properties.

Contribution

It introduces a new approach to approximate weak geodesics globally, enabling detailed analysis of the Mabuchi energy's properties in Kähler geometry.

Findings

Established convexity of Mabuchi functional along weak geodesics

Demonstrated continuity properties of the Mabuchi energy

Developed a method for global approximation using Monge-Ampère equations

Abstract

We are analysing the convexity and continuity properties of the Mabuchi functional along weak geodesics. The key technical point in our paper is the global approximation of weak geodesics obtained via a well-chosen family of Monge-Amp\`ere equations.