The volume-preserving mean curvature flow in Euclidean space
type: Item
instance of: scholarly article
title: The volume-preserving mean curvature flow in Euclidean space
zbMATH Open document ID: 1182.53061
author: Haozhao Li
published in: Pacific Journal of Mathematics
publication date: 2009-11-12T00:00:00Z
review text: The author proves a convergence of the volume-preserving mean curvature flow of hypersurfaces in Euclidean space under some initial integral pinching conditions on the second fundamental form. Let \(M\) be a closed oriented immersed surface in \({\mathbb R}^{n+1}\) given locally by an immersion \(F_0\). Let \(F_t(x)\) be a flow satisfying \[ \partial _t F_t(x) = (h-H)\nu(x,t),\quad F_t(x)|_{t=0} = f_0(x), \] where \(\nu(x,t)\) is the outer unit normal vector on \(M_t = F_t(M)\), \(H(x,t)\) is the mean curvature, and \(h(t)\) is its average defined by \(h(t) = \frac{1}{V_t}\int H(x,t)\, dv_t(x)\). Here, \(V_t\) denotes the volume of \(M_t\) with respect to the induced metric. The main result is the following. Given positive real numbers \(v, \Lambda, \gamma\), there exists a positive real number \(\varepsilon = \varepsilon(v, \Lambda, \gamma, n)>0\) such that if \(M^n\) is a closed oriented immersed surface in \({\mathbb R}^{n+1}\) satisfying \[ {\text{vol}}(M) \leq v,\quad |A| \leq \Lambda,\quad h \geq \gamma,\quad \int_M |{\AA}|^2 \leq \varepsilon,\tag{1} \] then the volume-preserving mean curvature flow with the initial surface \(M\) converges exponentially fast to a round sphere. Here, \({\AA} = A - \frac{H}{n} g\) is the traceless part of the second fundamental form \(=A\), and \(g\) is the induced metric. The condition (1) implies that both \(|\AA|\) and \(|H-h|\) are small after a short time and so decay exponentially in a short time provide that the average of the mean curvature is strictly positive. On the other hand, that they decay exponentially implies that they are both small in the next time interval. Finally, using iteration arguments, long time existence and exponential decay for all time follow. The condition on \(A\) can be weakened in the case \(n=2\). Namely, for given positive constants \(v_0, v_1, \Lambda, \gamma\) such that \(M^2 \subset {\mathbb R}^3\) is a compact smoothly immersed surface satisfying \[ v_0 < {\text{vol}}(M) < v,\quad \gamma …
reviewed by: Gabjin Yun
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 5632309
zbMATH Keywords: volume-preserving mean curvature flow
zbMATH Keywords: stability
zbMATH Keywords: second fundamental form
zbMATH Keywords: hypersurface
MaRDI profile type: Publication
full work available at URL: pjm.2009.243.331
OpenAlex ID: W1988427325
Wikidata QID: Q126089535
DOI: 10.2140/PJM.2009.243.331
description: scientific article; zbMATH DE number 5632309