Approximation of weak geodesics and subharmonicity of Mabuchi energy, II: $\epsilon$-geodesics

TL;DR

This paper investigates the convexity of the Mabuchi functional along certain geodesics using $psilon$-geodesics, proving convergence and non-degeneracy properties relevant to K"ahler geometry.

Contribution

It establishes $L^2$-convergence of fiberwise volume elements and uniform fiberwise non-degeneracy of geodesics under $psilon$-affine conditions, advancing understanding of Mabuchi energy.

Findings

Proved $L^2$-convergence of fiberwise volume elements.

Established uniform fiberwise non-degeneracy of geodesics.

Demonstrated strict convexity of Mabuchi functional along $C^{1,1}$-geodesics.

Abstract

The purpose of this article is to study the strict convexity of the Mabuchi functional along a $C^{1,1}$-geodesic, with the aid of the $\epsilon$-geodesics. We proved the $L^2$-convergence of the fiberwise volume element of the $\epsilon$-geodesic. Moreover, the geodesic is proved to be uniformly fiberwise non-degenerate if the Mabuchi functional is $\epsilon$-affine.