The heat flow and harmonic maps between complete manifolds
type: Item
instance of: scholarly article
title: The heat flow and harmonic maps between complete manifolds
zbMATH Open document ID: 1001.58011
DOI: 10.1007/BF02921799
author: Jiaping Wang
published in: The Journal of Geometric Analysis
publication date: 2000-06-22T00:00:00Z
review text: This paper consists of two parts. In the first one, the author extends the result of Li and Tam for heat equations of harmonic maps between complete noncompact manifolds. Let \(M\), \(N\) be complete noncompact Riemannian manifolds, and let \(\varphi\) be a \(C^1\)-map from \(M\) into \(N\). Consider the initial value problem for heat equations of harmonic maps: \(\frac{\partial u}{\partial t}(x,t) = \tau(u)(x,t)\), \(u(x,0) = \varphi(x)\) for \(x \in M\) and \(0 \leq t < \infty\), where \(\tau (u)\) is the tension field of \(u\). Li and Tam showed long-time existence of a unique solution for the above problem when the Ricci curvature \(\text{Ric}_M \geq -C\), the sectional curvature \(K_N \leq 0\), the map \(\varphi\) is bounded and \(\sup_M e(\varphi) < \infty\), where \(C\) is a nonnegative constant, and \(e(\varphi)\) is the energy density of the map \(\varphi\). The author proves a similar result under weaker conditions \(\text{Ric}_M \geq -C( 1+r(x))^{2-\varepsilon}\), \(K_N \leq 0\) and \(\sup_M e(\varphi) (x) \leq C \exp(C r(x)^{2-\varepsilon})\), where \(r(x)\) is the distance between \(x\) and the fixed point \(x_0\). He also solves the problem with some assumptions on tension fields of the initial map \(\varphi\). In the second part of this paper, the author constructs a solution of the Dirichlet problem at infinity for proper harmonic maps between hyperbolic spaces with the boundary map having finitely many points at which the energy density vanishes or the map being not differentiable.
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 35K99
zbMATH DE Number: 1463782
zbMATH Keywords: harmonic map
zbMATH Keywords: heat flow
zbMATH Keywords: complete manifold
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: global solution
reviewed by: Nobumitsu Nakauchi
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02921799
OpenAlex ID: W2143684112
description: scientific article; zbMATH DE number 1463782