Evolving compact locally convex curves and convex hypersurfaces
type: Item
instance of: scholarly article
title: Evolving compact locally convex curves and convex hypersurfaces
zbMATH Open document ID: 1486.52002
author: Laiyuan Gao
author: Yun-tao Zhang
published in: Manuscripta Mathematica
publication date: 2022-01-14T00:00:00Z
review text: Let \(M^n\) be a compact, smooth and locally convex hypersurface in the (\(n+1\))-dimensional Euclidean space \(\mathbb{E}^{n+1}\), and let \(X\) be a smooth immersion of \(M^n\) into \(\mathbb{E}^{n+1}\). In this paper, the authors consider the next flow of locally convex surfaces: \[ \left\{ \begin{array}{ll} \frac{\partial X}{\partial t}(x,t)=\bigl[R(x,t)+W(t)-(n+1)S(x,t)\bigr]N(x,t) & \text{ on }M^n\times(0,\omega),\\ X(x,0)=X_0(x) & \text{ on }M^n, \end{array} \right. \] where \(R\) represents the sum of the radii of curvature, \(S\) denotes the support function of the surface, \(W\) is its mean width and \(N\) is the outer normal vector. Then they prove that if \(M_0^n\) is an \(n\)-manifold immersed in \(\mathbb{E}^{n+1}\), the above flow with initial \(M_0^n\) exists globally and deforms the evolving curve into an \(m\)-fold circle in the plane if \(n=1\), and drives the evolving hypersurface into a standard sphere if \(n>1\), as \(t\to\infty\).
reviewed by: Maria A. Hernández Cifre
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 51M05
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 7457117
zbMATH Keywords: curvature flow
zbMATH Keywords: compact locally convex hypersurfaces
MaRDI profile type: Publication
full work available at URL: s00229-021-01278-7
OpenAlex ID: W3130116063
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DOI: 10.1007/S00229-021-01278-7
description: scientific article; zbMATH DE number 7457117