Paper · Dec 4, 2022

Flow by Gauss curvature to the -Gaussian Minkowski problem

Weimin Sheng, Ke Xue

math.DGWider flowsarXiv:2212.01822
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Abstract

In this paper, we study the -Gaussian Minkowski problem, which arises in the -Brunn-Minkowski theory in Gaussian probability space. We use Aleksandrov's variational method with Lagrange multipliers to prove the existence of the logarithmic Gauss Minkowski problem. We construct a suitable Gauss curvature flow of closed, convex hypersurfaces in the Euclidean space , and prove its long-time existence and converges smoothly to a smooth solution of the normalized Gaussian Minkowski problem in cases of and with even prescribed function respectively. We also provide a parabolic proof in the smooth category to the -Gaussian Minkowski problem in cases of and with even prescribed function, respectively.

Comments: This is a revised version of an early paper. arXiv admin note: text overlap with arXiv:1712.07774; text overlap with arXiv:2103.00189 by other authors

Other curvature flowsGauss curvature flowLong-time existence & convergenceGeometric inequalities

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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) 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) Geometric inequalities Sharp inequalities between area, volume and integrals of curvature, with equality only for round spheres. Start with: Mean curvature flow and geometric inequalities (Topping, 1998)