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

Chapter in book
Authors Robert Berman
Published in Analysis Meets Geometry. (Trends in Mathematics). Andersson M., Boman J., Kiselman C., Kurasov P., Sigurdsson R. (eds)
Pages 111-120
ISBN 978-3-319-52471-9
ISSN 2297-0215
Publisher Springer
Publication year 2017
Published at Department of Mathematical Sciences
Pages 111-120
Language en
Subject categories Algebra and geometry


© 2017 Springer International Publishing. 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 C 1,1 -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 ℂ n is given.

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