Curvature bound for Lp Minkowski problem

We establish curvature estimates for anisotropic Gauss curvature flows. By using this, we show that given a measure μ with a positive smooth density f, any solution to the Lp Minkowski problem in Rn+1 with p≤−n+2 is a hypersurface of class C1,1. This is a sharp result because for each p∈[−n+2,1) the...

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Bibliographic Details
Published inAdvances in mathematics (New York. 1965) Vol. 458
Main Authors Choi, Kyeongsu, Kim, Minhyun, Lee, Taehun
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.12.2024
Subjects
Curvature flow
Minkowski problem
Regularity estimates
secondary
Regularity estimates
Curvature flow
Minkowski problem
primary
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Summary:We establish curvature estimates for anisotropic Gauss curvature flows. By using this, we show that given a measure μ with a positive smooth density f, any solution to the Lp Minkowski problem in Rn+1 with p≤−n+2 is a hypersurface of class C1,1. This is a sharp result because for each p∈[−n+2,1) there exists a convex hypersurface of class C1,1n+p−1 which is a solution to the Lp Minkowski problem for a positive smooth density f. In particular, the C1,1 regularity is optimal in the case p=−n+2 which includes the logarithmic Minkowski problem in R3.
ISSN:0001-8708
DOI:10.1016/j.aim.2024.109959