A bridge principle for harmonic diffeomorphisms between surfaces
type: Item
instance of: scholarly article
title: A bridge principle for harmonic diffeomorphisms between surfaces
zbMATH Open document ID: 1023.53049
DOI: 10.1023/A:1014744428228
author: Derchyi Wu
author: Yng-Ing Lee
author: Ai-Nung Wang
published in: Annals of Global Analysis and Geometry
publication date: 2002-06-09T00:00:00Z
review text: The authors prove the following result. Theorem. Let \(\Sigma_\iota\) be embedded closed surfaces of a Riemannian manifold \(M\), \(\tilde\Sigma_\iota\) be embedded closed surfaces of a Riemannian manifold \(\tilde M\), \(\dim M,\dim\tilde M\geq 3\), and \(\text{genus}(\Sigma_\iota)=\text{genus}(\tilde\Sigma_\iota)\geq 2\), for \(\iota=0,1\). Let \(f_\iota:\Sigma_\iota\to\tilde\Sigma_\iota\) be harmonic diffeomorphisms. Suppose that \(\gamma\), \(\tilde\gamma\) are Jordan arcs connecting \(\Sigma_0\) and \(\Sigma_1\), \(\tilde\Sigma_0\) and \(\tilde\Sigma_1\), respectively, satisfying that \(\gamma\cap\Sigma_\iota=\gamma(\iota)\), and \(\tilde\gamma\cap\tilde\Sigma_\iota=\tilde\gamma(\iota)=f_\iota(\gamma(\iota))\), for \(\iota=0,1\). Then in any sufficiently small \(\epsilon\)-neighborhood of \(\gamma\) (small \(\epsilon\)-neighborhood of \(\tilde\gamma\), respectively), one can connect \(\Sigma_0\) and \(\Sigma_1\) by a bridge tube \(T_\epsilon\), (\(\tilde\Sigma_0\) and \(\tilde\Sigma_1\) by a bridge tube \(\tilde T_\epsilon\), respectively), and find a harmonic diffeomorphism \(F_\epsilon\) from the smooth bridged surface \(\Sigma_0\sharp_{T_\epsilon}\Sigma_1\) onto the smooth bridged surface \(\tilde\Sigma_0\sharp_{\tilde T_\epsilon}\tilde\Sigma_1\). Furthermore, as the radius of \(T_\epsilon\) shrinks to zero (i.e., \(\epsilon\to 0\)), \(\{F_\epsilon\}\) converges to \(f_\iota\) (in the \(C^k\)-topology, \(k>2\)) on each compact subset of \(\Sigma_\iota\setminus\gamma_\iota\), for \(\iota=0,1\). As the authors write, the main point is the limit behavior. There exist corresponding results in the higher dimensional case, but these methods break down in the situation considered here.
Mathematics Subject Classification ID: 53C43
Mathematics Subject Classification ID: 58E20
zbMATH DE Number: 1752564
zbMATH Keywords: bridge principle
zbMATH Keywords: energy
zbMATH Keywords: harmonic diffeomorphism
zbMATH Keywords: minimal submanifold
reviewed by: Linus Kramer
MaRDI profile type: Publication
full work available at URL: a:1014744428228
OpenAlex ID: W170027074
description: scientific article; zbMATH DE number 1752564