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arXiv · 1405.6482

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

Robert J. Berman
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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 \C^{n} is given.

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Robert J. Berman. 2014-05-26. On the optimal regularity of weak geodesics in the space of metrics on a polarized manifold. https://arxiv.org/abs/1405.6482

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