On the optimal regularity of weak geodesics in the space of metrics on a polarized manifold

Abstract

Let (X,L) be a polarized compact manifold, i.e., L is an ample line bundle over X and denote by ℋ the infinite-dimensional space of all positively curved Hermitian metrics on L equipped with the Mabuchi metric. In this short note we show, using Bedford–Taylor type envelope techniques developed in the authors previous work [3], that Chen’s weak geodesic connecting any two elements in ℋ are C1,1-smooth, i.e., the real Hessian is bounded, for any fixed time t, thus improving the original bound on the Laplacians due to Chen. This also gives a partial generalization of Blocki’s refinement of Chen’s regularity result. More generally, a regularity result for complex Monge–Ampère equations over X × D, for D a pseudoconvex domain in ℂn is given.