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.

arXiv commentWe corrected some errors and typos, and added several remarks and more details to explain this paper. Moreover, the more complicated proof of the strong L^2-convergence of the fiberwise volume element of the epsilon geodesic has been moved to the Appendix, and only remain the simpler one in the main context

References in corpus (2)

• On deformation of extremal metrics
• Geodesic Convexity of Small Neighborhood in the Space of Kähler Potentials