Behaviour of singularities of the rotationally symmetric, volume-preserving mean curvature flow
type: Item
instance of: scholarly article
title: Behaviour of singularities of the rotationally symmetric, volume-preserving mean curvature flow
zbMATH Open document ID: 1045.53045
published in: Calculus of Variations and Partial Differential Equations
publication date: 2003-08-21T00:00:00Z
review text: Let \(x_{t}:M^{n}\rightarrow \mathbb{R}^{n+1}\) be a one-parameter family of smooth immersions and \(M_{t}=x_{t}(M^{n}).\) In the case when \(x_{t}\) satisfies the evolution equation \(\frac{d}{dt}x(p,t)=-H(p,t)\nu (p,t),\) \((t>0),\) where \( \nu (p,t)\) denotes a choice of unit normal to \(M_{t}\) at \(x(p,t)\), \(H(p,t)\) denotes the mean curvature with respect to this normal, it is said that the hypersurfaces \(M_{t}\) move by mean curvature. The mean curvature flow was studied by \textit{K. A. Brakke} [The motion of a surface by its mean curvature. (Princeton, New Jersey) (1978; Zbl 0386.53047)]. For compact, convex initial surfaces without boundary, \textit{G. Huisken} [J. Differ. Geom. 20, 237--266 (1984; Zbl 0556.53001)] proved that they converge asymptotically to round spheres. The question of the formation of singularities in the nonconvex case was analyzed by \textit{G. Huisken} [J. Differ. Geom. 31, No. 1, 285--299 (1990; Zbl 0694.53005)], \textit{M. A. Grayson} [Duke Math. J. 58, No. 3, 555--558 (1989; Zbl 0677.53059)], \textit{G. Dziuk} and \textit{B. Kawohl} [J. Differ. Equations 93, No. 1, 142--149 (1991; Zbl 0749.53001)]. The author is interested in the evolution of compact hypersurfaces \(M_{t}\) enclosing a prescribed constant volume \(V\). This means that \(x_{t}\) satisfies the following evolution equation: \(\frac{d}{dt}x(p,t)= -(H(p,t)-h(t))\nu (p,t),\) \((t>0)\), where \(h(t)=( \int_{M}\,dg_{t}) ^{-1}\cdot \int_{M}H \,dg_{t}\) is the average of the mean curvature, and \(g_{t}\) denotes the metric on \(M_{t}\). The initial hypersurface \(M_{0}\) is supposed to be smoothly embedded in the domain \(G=\{x\in R^{n+1}:0<x_{n+1}<d\},\) \((d>0)\) and \(\partial M_{0}\subset \partial G.\) The surface areas of \(M_{t}\) are decreasing under this flow, while the enclosed volume is constant. In this case, the hypersurfaces can be expected to converge to a surface of constant mean curvature which solves the isoperimetric problem. The author imposes volume…
reviewed by: Ilie Burdujan
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 35F25
zbMATH DE Number: 1969049
zbMATH Keywords: mean curvature flow
zbMATH Keywords: asymptotic behavior of singularities
zbMATH Keywords: volume constraints
author: Maria Athanassenas
MaRDI profile type: Publication
full work available at URL: s00526-002-0098-4
OpenAlex ID: W1969387108
DOI: 10.1007/S00526-002-0098-4
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description: scientific article; zbMATH DE number 1969049