Paper · Oct 12, 2022

Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space

Yong Wei, Bo Yang, Tailong Zhou

math.DGWider flowsarXiv:2210.06035v2Trans. Amer. Math. Soc. 377 (2024), no. 4, 2821-2854
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Abstract

We consider the volume preserving flow of smooth, closed and convex hypersurfaces in the hyperbolic space with the speed given by arbitrary positive power of the Gauss curvature. We prove that if the initial hypersurface is convex, then the smooth solution of the flow remains convex and exists for all positive time . Moreover, we apply a result of Kohlmann which characterises the geodesic ball using the hyperbolic curvature measures and an argument of Alexandrov reflection to prove that the flow converges to a geodesic sphere exponentially in the smooth topology. This can be viewed as the first result for non-local type volume preserving curvature flows for hypersurfaces in the hyperbolic space with only convexity required on the initial data.

Comments: v2, 29 pages, 3 figures, minor revision in the proof of Lemma 6.6

Other curvature flowsGauss curvature flowVolume-preserving & constrained 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) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986) 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)