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.
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Berman, R.J. (2017). On the Optimal Regularity of Weak Geodesics in the Space of Metrics on a Polarized Manifold. In: Andersson, M., Boman, J., Kiselman, C., Kurasov, P., Sigurdsson, R. (eds) Analysis Meets Geometry. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-52471-9_7
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DOI: https://doi.org/10.1007/978-3-319-52471-9_7
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Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-52469-6
Online ISBN: 978-3-319-52471-9
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