Liouville theorem for harmonic maps with potential
type: Item
instance of: scholarly article
title: Liouville theorem for harmonic maps with potential
zbMATH Open document ID: 0911.58008
author: Qun Chen
published in: Manuscripta Mathematica
publication date: 1998-08-02T00:00:00Z
full work available at URL: https://eudml.org/doc/156391
review text: Let \(M\), \(N\) be complete Riemannian manifolds, and let \(H\) be a function on \(N\). A harmonic map with potential \(H\) is a map \(u: M\to N\) that is a critical point of the energy functional \[ E_H(u)=\int_M(e(u)-H(u)), \] where \(e\) is the energy density. Assume that \(M\) is a complete noncompact manifold with only one end, and that there is a nonincreasing function \(k\geq 0\) with \(\int_0^\infty t k(t) dt<+\infty\) such that the sectional curvature of \(M\) is bounded below by \(-k(d(\cdot,x))\) for some point \(x\in M\). Let the sectional curvature of \(N\) be bounded above by some \(K\geq 0\), and let \(B_q(\tau)\) be a ball of radius \(\tau<\pi/2\sqrt K\), lying inside the cut locus of \(q\). Finally, assume that \({\partial H\over\partial\rho}\leq 0\), where \(\rho\) denotes the distance from \(q\). The author proves that then any harmonic map \(u: M\to B_q(\tau)\) with potential \(H\) is constant, and that \(u(M)\) is a critical point of \(H\). On the other hand, for \(M\) having more than two ends, the existence of nonconstant harmonic maps \(u: M\to B_q(\tau)\) is known.
reviewed by: Sebastian Goette
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 58J60
zbMATH DE Number: 1158115
zbMATH Keywords: Liouville theorem
zbMATH Keywords: harmonic maps with potential
MaRDI profile type: Publication
OpenAlex ID: W2047221398
DOI: 10.1007/S002290050044
description: scientific article; zbMATH DE number 1158115