Paper · Sep 10, 2022

A class of anisotropic inverse Gauss curvature flows and dual Orlicz Minkowski type problem

Shanwei Ding, Guanghan Li

math.APWider flowsarXiv:2209.04601
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Abstract

In this paper, we study the long-time existence and asymptotic behavior for a class of anisotropic inverse Gauss curvature flows. By the stationary solutions of anisotropic flows, we obtain some new existence results for the dual Orlicz Minkowski type problem and even dual Orlicz Minkowski type problem for smooth measures, which is the most reasonable extension of the dual Minkowski problem from the dual point of view. The results of corresponding versions are dual Minkowski problem for ; and even dual Minkowski problem for , or , or some ranges of , which contain all existence results for smooth measures up to now except or ( Minkowski problem).

Comments: arXiv admin note: text overlap with arXiv:2207.03114

Other curvature flowsGauss curvature flowAnisotropic & nonlocal flowsLong-time existence & convergence

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)