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