Paper · Apr 23, 2023

Curvature bound for Minkowski problem

Kyeongsu Choi, Minhyun Kim, Taehun Lee

math.DGWider flowsarXiv:2304.11617v3Adv. Math. 458 (2024), part A, Paper No. 109959, 31 pp
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Abstract

We establish curvature estimates for anisotropic Gauss curvature flows. By using this, we show that given a measure with a positive smooth density , any solution to the Minkowski problem in with is a hypersurface of class . This is a sharp result because for each there exists a convex hypersurface of class which is a solution to the Minkowski problem for a positive smooth density . In particular, the regularity is optimal in the case which includes the logarithmic Minkowski problem in .

Comments: 25 pages, to appear in Adv. Math

Other curvature flowsGauss curvature flowAnisotropic & nonlocal flows

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Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Gauss curvature flow Wider flows Convex hypersurfaces shrink with normal speed a power K^α of their Gauss curvature. Firey proposed it as a model of stones worn by waves; for α>… Start with: Shapes of worn stones (Firey, 1974) Anisotropic & nonlocal flows Wider flows When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere. Start with: Crystalline variational problems (Taylor, 1978)