Heat flow for extrinsic biharmonic maps with small initial energy
type: Item
instance of: scholarly article
title: Heat flow for extrinsic biharmonic maps with small initial energy
zbMATH Open document ID: 1080.58017
DOI: 10.1023/B:AGAG.0000047526.21237.04
author: Tobias Lamm
published in: Annals of Global Analysis and Geometry
publication date: 2005-04-18T00:00:00Z
review text: For mappings between compact Riemannian manifolds \(M\) and \(N\subseteq \mathbb{R}^k\), the \(L^2\) gradient flow for the extrinsic bi-energy \(F(u):={1\over2}\int_M| \Delta u| ^2\,dx\) is considered, where \(\Delta\) is applied component-wise in \(\mathbb{R}^k\). If \(\dim M\leq3\) or if \(\dim M=4\) and the initial bi-energy \(F(u_0)\) is small, it is proved that the extrinsic biharmonic map heat flow exists for all times. Moreover, for general initial data in the case \(\dim M=4\), possible singularity formation is understood completely; it can only happen by ``bubbling off'' of bi-energy in some sort of bi-harmonic bubble. The methods involved are a careful adaption to higher order of the corresponding proof by Struwe for the harmonic map heat flow [\textit{M. Struwe}, Comment. Math. Helv. 60, 558--581 (1985; Zbl 0595.58013)], combined with interpolation inequalities influenced by [\textit{E. Kuwert} and \textit{R. Schätzle}, J. Differ. Geom. 57, 409--441 (2001; Zbl 1035.53092)].
reviewed by: Andreas Gastel
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 58J35
zbMATH DE Number: 2158450
zbMATH Keywords: biharmonic maps
zbMATH Keywords: heat flow
zbMATH Keywords: existence
zbMATH Keywords: regularity
zbMATH Keywords: bubbling
zbMATH Keywords: initial value problem
MaRDI profile type: Publication
full work available at URL: b:agag.0000047526.21237.04
OpenAlex ID: W2091738437
description: scientific article; zbMATH DE number 2158450