Paper · Nov 19, 2021

On a Class of Fully Nonlinear Curvature Flows in Hyperbolic Space

Fang Hong

math.DGWider flowsarXiv:2111.10170
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Abstract

In this paper, we study a class of flows of closed, star-shaped hypersurfaces in hyperbolic space with speed , where is the -th elementary symmetric polynomial of the principal curvatures, , are positive constants and is the distance from points on the hypersurface to the origin. We obtain convergence results under some assumptions of , and . When , and the initial hypersurface is mean convex, we prove that the mean convex solution to the flow for exists for all time and converges smoothly to a sphere. When , and the initial hypersurface is uniformly convex, we prove that the uniformly convex solution to the flow exists for all time and converges smoothly to a sphere. In particular, we generalize Li-Sheng-Wang's results from Euclidean space to hyperbolic space.

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Submitted
Nov 19, 2021
Subjects
math.DG
Journal
J Geom Anal 36, 55 (2026)
DOI
10.1007/s12220-025-02254-3

Comments: 25 pages

Other curvature flowsLong-time existence & convergence

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Read around it

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) 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)