Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space

Abstract

We consider the dynamic property of the volume preserving mean curvature flow. This flow was introduced by Huisken (J Reine Angew Math 382:35–48, 1987) who also proved it converges to a round sphere of the same enclosed volume if the initial hypersurface is strictly convex in Euclidean space. We study the stability of this flow in hyperbolic space. In particular, we prove that if the initial hypersurface is hyperbolically mean convex and close to an umbilical sphere in the

L^2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^2$$\end{document}

-sense, then the flow exists for all time and converges exponentially to an umbilical sphere.

Keywords

Volume preserving mean curvature flow / h-mean convex in hyperbolic space / Dynamical stability

References

[1] AndrewsB, ChenX, WeiY. Volume preserving flow and Alexandrov–Fenchel type inequalities in hyperbolic space. J. Eur. Math. Soc., 2021, 2372467-2509.

[2] AubinTSome Nonlinear Problems in Riemannian Geometry. Springer Monographs in Mathematics, 1998BerlinSpringer.

[3] BertiniM, PipoliG. Volume preserving non-homogeneous mean curvature flow in hyperbolic space. Differ. Geom. Appl., 2017, 54part B448-463.

[4] Cabezas-RivasE, MiquelV. Volume preserving mean curvature flow in the hyperbolic space. Indiana Univ. Math. J., 2007, 5652061-2086.

[5] Cabezas-RivasE, MiquelV. Volume preserving mean curvature flow of revolution hypersurfaces between two equidistants. Calc. Var. Partial Differ. Equ., 2012, 431–2185-210.

[6] ChowB, PengL, NiLHamilton’s Ricci Flow, 2006ProvidenceAmerican Mathematical Society77

[7] De LellisC, MüllerS. Optimal rigidity estimates for nearly umbilical surfaces. J. Differ. Geom., 2005, 69175-110.

[8] EichmairM, MetzgerJ. On large volume preserving stable CMC surfaces in initial data sets. J. Differ. Geom., 2012, 91181-102.

[9] EscherJ, SimonettG. The volume preserving mean curvature flow near spheres. Proc. Am. Math. Soc., 1998, 12692789-2796.

[10] GuoS, LiG, WuC. Volume-preserving flow by powers of the $m$-th mean curvature in the hyperbolic space. Commun. Anal. Geom., 2017, 252321-372.

[11] HamiltonRS. Three-manifolds with positive Ricci curvature. J. Differ. Geom., 1982, 172255-306.

[12] HoffmanD, SpruckJ. Sobolev and isoperimetric inequalities for Riemannian submanifolds. Commun. Pure Appl. Math., 1974, 27: 715-727.

[13] HuY, LiH, WeiY, ZhouT. Contraction of surfaces in hyperbolic space and in sphere. Calc. Var. Partial Differ. Equ., 2020, 595172.

[14] HuangZ, LinL. Stability of the surface area preserving mean curvature flow in Euclidean space. J. Geol., 2015, 1063483-501

[15] HuiskenG. Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom., 1984, 201237-266.

[16] HuiskenG. Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature. Invent. Math., 1986, 843463-480.

[17] HuiskenG. The volume preserving mean curvature flow. J. Reine Angew. Math., 1987, 382: 35-48

[18] HuiskenG, SinestrariC. Mean curvature flow singularities for mean convex surfaces. Calc. Var. Partial Differ. Equ., 1999, 811-14.

[19] HuiskenG, YauS-T. Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature. Invent. Math., 1996, 1241–3281-311.

[20] LiH. The volume-preserving mean curvature flow in Euclidean space. Pac. J. Math., 2009, 2432331-355.

[21] McCoyJ. The surface area preserving mean curvature flow. Asian J. Math., 2003, 717-30.

[22] McCoyJ. The mixed volume preserving mean curvature flow. Math. Z., 2004, 2461–2155-166.

[23] MichaelJH, SimonLM. Sobolev and mean-value inequalities on generalized submanifolds of $R^{n}$. Commun. Pure Appl. Math., 1973, 26: 361-379.

[24] SpivakMA Comprehensive Introduction to Differential Geometry, 19792WilmingtonPublish or PerishIV

[25] ToppingP. Relating diameter and mean curvature for submanifolds of Euclidean space. Comment. Math. Helv., 2008, 833539-546.

[26] WangG, XiaC. Isoperimetric type problems and Alexandrov–Fenchel type inequalities in the hyperbolic space. Adv. Math., 2014, 259: 532-556.

[27] WuJ-Y, ZhengY. Relating diameter and mean curvature for Riemannian submanifolds. Proc. Am. Math. Soc., 2011, 139114097-4104.

[28] XuH, LengY, ZhaoE. Volume-preserving mean curvature flow of hypersurfaces in space forms. Int. J. Math., 2014, 253, 1450021.

[29] YeR. Ricci flow, Einstein metrics and space forms. Trans. Am. Math. Soc., 1993, 3382871-896.

RIGHTS & PERMISSIONS

Peking University