Gradient flow for the Willmore functional
type: Item
instance of: scholarly article
title: Gradient flow for the Willmore functional
zbMATH Open document ID: 1029.53082
published in: Communications in Analysis and Geometry
publication date: 2003-04-07T00:00:00Z
review text: The Willmore energy of a closed, immersed surface \(f:\Sigma\to \mathbb{R}^n\) with induced area measure \(d\mu\) is given by \(W(f)=\int_\Sigma|A^0 |^2d\mu\), where \(A^0=A-\frac 12 g \otimes H\) is the tracefree part of the second fundamental form \(A\). It is not known whether there can develop singularities under the Willmore flow (related to the above energy) in finite time. One of the main results of the paper under review is to obtain a lower bound on the lifespan which depends only on the concentration of curvature for the intial surface. The main theorem is the following: Theorem: Let \(f_0:\Sigma \to\mathbb{R}^n\) be a smooth immersion. There are constants \(\varepsilon_0>0\), \(c<\infty\) depending only on \(n\), such that if \(\rho>0\) is chosen with \[ \int_{B_\rho(x)} |A_0 |^2 d\mu_0\leq \varepsilon\leq \varepsilon_0 \] for any \(x\in\mathbb{R}^n\), then the maximal time \(T\) of smooth existence for the Willmore flow with initial data \(f_0\) satisfies \(T\geq {1\over c}\rho^4\), and one has the estimate \[ \int_{B_\rho (x)}|A|^2 d\mu\leq c\varepsilon \] for \(0\leq t\leq {1\over c}\rho^4\).
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 35K10
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 1893848
zbMATH Keywords: gradient flow
zbMATH Keywords: Willmore flow
zbMATH Keywords: Willmore energy
author: Ernst Kuwert
author: Reiner Schätzle
reviewed by: Andrzej Szczepański
MaRDI profile type: Publication
DOI: 10.4310/CAG.2002.V10.N2.A4
description: scientific article; zbMATH DE number 1893848