On the optimal regularity of weak geodesics in the space of metrics on a polarized manifold
Kapitel i bok, 2017

Let (X,L) be a polarized compact manifold, i.e., L is an ample line bundle over X and denote by ? 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 ? 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 ?n is given.


Robert Berman

Chalmers, Matematiska vetenskaper, Algebra och geometri

Göteborgs universitet

Trends in Mathematics

22970215 (ISSN) 2297024X (eISSN)

9,78332E+12 111-120





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