Harmonic maps into spaces with an upper curvature bound in the sense of Alexandrov
type: Item
instance of: scholarly article
title: Harmonic maps into spaces with an upper curvature bound in the sense of Alexandrov
zbMATH Open document ID: 1044.58021
author: Chikako Mese
published in: Mathematische Zeitschrift
publication date: 2004-06-17T00:00:00Z
review text: The author studies harmonic maps into metric spaces satisfying the CAT(\(\kappa\)) comparison inequality (which are called here spaces of curvature bounded from above by \(\kappa\)). He develops the notion of order function for energy minimizing maps into such spaces and he generalizes several results which hold for energy minimizing maps into spaces with nonpositive curvature to this setting of spaces satisfying an upper curvature bound. The main results are the following: Theorem: let \((M,g)\) be a compact Riemannian manifold with or without boundary and let \((d_i)\) be a sequence of distance functions on a space \(X\) with curvature bounded from above by \(\kappa\). Assume that \(X\) is compact with respect to the topology induced by each \(d_i\), let \(h:M\to X\) be a continuous map and let \(f_i:M\to (X,d_i)\) be continuous energy minimizing maps in the homotopy class of \(h\) with \(f_i\) coinciding with \(h\) on \(\partial M\) if \(\partial M\not= \emptyset\). Let \(\delta_i\) be the pull-back distance of \(d_i\) under \(f_i\). If the energy of \(f_i\) is bounded from above by some constant that does not depend on \(i\) and if \(d_i\) converges uniformly to a distance function \(d_0\), then there exists a subsequence \((i'_i)\) of \((i)\) and an energy minimizing map \(f_0\) with respect to \(d_0\) so that \(f_{i'}\) converges pointwise to \(f\), \(\delta_{i'}(.,.)\) converges uniformly to \(d_0(f(.),f(.))\) and the energy of \(f_{i'}\) converges to that of \(f_0\). The result is an adaptation of a compactness theorem of \textit{N. J. Korevaar} and \textit{R. M. Schoen} [Commun. Anal. Geom. 5, No. 2, 333--387 (1997; Zbl 0908.58007)]. As an application, the author obtains the following: Theorem: Let \(S\) be a compact topological surface and let \(d\) be a distance function on \(S\) that makes it a metric space of curvature bounded from above by \(\kappa\), with the topology induced by the metric \(d\) being the topology of \(S\) as a surface. Let \(\Sigma_1…
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 53C45
Mathematics Subject Classification ID: 53C23
zbMATH DE Number: 2075747
zbMATH Keywords: harmonic maps
zbMATH Keywords: Alexandrov surfaces
zbMATH Keywords: comparison inequality
zbMATH Keywords: energy
zbMATH Keywords: energy minimizing map
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
full work available at URL: s002090100372
OpenAlex ID: W1982265497
DOI: 10.1007/S002090100372
description: scientific article; zbMATH DE number 2075747