Higher regularity of the inverse mean curvature flow
type: Item
instance of: scholarly article
title: Higher regularity of the inverse mean curvature flow
zbMATH Open document ID: 1161.53058
author: Gerhard Huisken
author: Tom Ilmanen
published in: Journal of Differential Geometry
publication date: 2009-02-09T00:00:00Z
review text: The authors study the inverse mean curvature flow (IMCF) dealing with a closed hypersurface \(F: M^n\times [0,T]\to \mathbb{R}^{n+1}\), whose evolution in time is governed by the equation \({\frac{dF}{dt} = \frac{1}{H} \nu}\), where \(H\) and \(\nu\) are the mean curvature and exterior unit normal of \(F\). It is known that if \(F_0=F(M^n,0)\) is a smooth star-shaped hypersurface of strictly positive mean curvature then IMCF has a smooth solution for all times which approaches a homothetically expanding spherical solution as \(t\to \infty\), see [\textit{C. Gerhardt}, J. Differ. Geom. 32, No. 1, 299--314 (1990; Zbl 0708.53045); \textit{J. I. E. Urbas}, Math. Z. 205, No. 3, 355--372 (1990; Zbl 0691.35048)]. On the other hand, if the initial hypersurface \(F_0\) is not star-shaped, then the singularities may develop; moreover it is proved by \textit{K.~Smoczyk} that in the case \(n=2\) such singularities can occur if the mean curvature \(H\) tends to zero somewhere during the evolution [Asian J. Math. 4, No. 2, 331--335 (2000; Zbl 0989.53040)]. The main result of the paper is the following lower bound for the mean curvature of star-shaped solutions to IMCF. Theorem. Let \(F: M^n\times [0,T]\to \mathbb{R}^{n+1}\) be a smooth star-shaped solution to IMCF. Suppose that the following estimates hold at \(t=0\): \[ 0<R_1\leq \langle F,\nu\rangle \leq R_2. \] Then there is a constant \(0<C_n<\infty\) depending only on \(n\) such that the estimates \[ \frac{1}{H\langle F,\nu\rangle}\leq C_n\max\left(\frac{1}{\sqrt{t}},1\right)\frac{1}{R_1}| F_0|^{1/n} \] and \[ H\geq \frac{1}{C_n}\min\left(\sqrt{t},1\right) e^{-t/n}\frac{R_1}{R_2}| F_0|^{-1/n} \] hold everywhere on \(M^n\times [0,T]\). Remark that if \(F\) is star-shaped and satisfies \(0<R_1\leq \langle F,\nu\rangle \leq R_2\) at \(t=0\), then during the evolution under IMCF it remains star-shaped and satisfies \(e^{t/n}R_1\leq \langle F,\nu\rangle \leq R_2e^{t/n}\) [cf Zbl 0708.53045]. Thus in the star-shaped setting \texts…
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 5504164
zbMATH Keywords: inverse mean curvature flow
zbMATH Keywords: star-shaped surface
zbMATH Keywords: weak solution
reviewed by: Vasyl Gorkaviy
Wikidata QID: Q115172959
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1226090483
OpenAlex ID: W2161845662
DOI: 10.4310/JDG/1226090483
description: scientific article; zbMATH DE number 5504164