The Willmore flow with small initial energy.
type: Item
instance of: scholarly article
title: The Willmore flow with small initial energy.
zbMATH Open document ID: 1035.53092
published in: Journal of Differential Geometry
publication date: 2002-08-15T00:00:00Z
review text: The authors consider the \(L^2\) gradient flow for the Willmore functional. In another paper, they have proved that the curvature concentrates if a singularity develops. Here they show that a suitable blowup converges to a nonumbilic (compact or noncompact) Willmore surface. Furthermore, an \(L^\infty\) estimate is derived for the tracefree part of the curvature of a Willmore surface, assuming that its \(L^2\) norm (the Willmore energy) is locally small. One consequence is that a properly immersed Willmore surface with restricted growth of the curvature at infinity and small total energy must be a plane or a sphere. Combining the results they obtain long time existence and convergence to a round sphere if the total energy is initially small.
Mathematics Subject Classification ID: 53C44
zbMATH DE Number: 1782684
zbMATH Keywords: Willmore surface
zbMATH Keywords: Willmore flow
zbMATH Keywords: Willmore functional
zbMATH Keywords: gradient flow
zbMATH Keywords: Willmore energy
Wikidata QID: Q115202779
author: Ernst Kuwert
author: Reiner Schätzle
reviewed by: Huafei Sun
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1090348128
OpenAlex ID: W1504258993
DOI: 10.4310/JDG/1090348128
description: scientific article; zbMATH DE number 1782684