Approximation of weak geodesics and subharmonicity of Mabuchi energy, II: $ε$-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 ε-geodesics. We proved the L 2 -convergence of the fiberwise volume element of the ε-geodesic. Moreover, the geodesic is proved to be uniformly fiberwise non-degenerate if the Mabuchi functional is ε-affine.