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

2014/05/26 by Robert J. Berman, Berman, Robert J.
Mathematics · #Algebraic Geometry and Number Theory #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.CV #math.DG

paper · pdf · doi:10.48550/arxiv.1405.6482

8 pages

arxiv created 2014/05/26 · arxiv updated 2014/05/27

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 [ber2], that Chen's weak geodesic connecting any two elements in H are C^1,1-smooth, i.e. the real Hessian is bounded, for any fixed time t, thus improving the original bound on the Laplacins 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-Ampere equations over X\timesD, for D a pseudconvex domain in \Cⁿ is given.

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