Harmonic mean curvature flow on surfaces of negative Gaussian curvature
type: Item
instance of: scholarly article
title: Harmonic mean curvature flow on surfaces of negative Gaussian curvature
zbMATH Open document ID: 1127.53058
author: Panagiota Daskalopoulos
author: Richard S. Hamilton
published in: Communications in Analysis and Geometry
publication date: 2007-04-10T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.cag/1175790873
review text: The \textit{harmonic mean curvature flow} is the geometric evolution equation \[ \frac{\partial P}{\partial t}=\frac{K}{H} \nu, \] in which each point of a surface evolves in the normal direction \(\nu\) with velocity equal to the quotient of the Gaussian curvature \(K\) by the mean curvature \(H\). The convex and weakly convex cases have been studied in [1]-[4]. In this paper, the authors consider a surface of revolution \(f(x,t)\) with boundary, whose principal eigenvalues have opposite signs, and whose mean curvature is negative at each point. The function \(f\) satisfies the fully nonlinear equation \[ \frac{\partial f}{\partial t}=\frac{f_{xx}}{-ff_{xx}+f_x^2+1}. \] The principal coefficient of the linearization is non-negative, independent of the signs of \(H\) and \(K\). Assume that the initial surface satisfies \(f >0\), \(f_{xx} >0\), and \(\tilde H := \;-f f_{xx}+f_x^2+1 <0\) on an interval \(I=[0,1]\), as well as the boundary growth condition \(c\leq x^{2-p}(1-x)^{2-p}f_{xx}\leq C,\;\;0 <x <1\) for some \(0 <p <1\), and constants \(c >0\) and \(C< \infty\). The main result is that there is a time \(T_0\) up to which there exists a unique solution of the flow, so that the mean curvature is negative for \(t< T_0\). Sharp a priori derivative bounds are derived to show smoothness of the flow and positivity of \(\tilde H\) up to \(T_0\). At \(T_0\), \(\tilde H\) vanishes on an interval whose length depends on the initial surface. The area of the surface is preserved under the flow, and the boundary of the surface moves by the curve shortening flow.
reviewed by: Christine Guenther
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 5141317
zbMATH Keywords: harmonic mean curvature flow
MaRDI profile type: Publication
OpenAlex ID: W1992719981
DOI: 10.4310/CAG.2006.V14.N5.A4
description: scientific article; zbMATH DE number 5141317