The Morse theory of two-dimensional closed branched minimal surfaces and their generic non-degeneracy in Riemannian manifolds
type: Item
instance of: scholarly article
title: The Morse theory of two-dimensional closed branched minimal surfaces and their generic non-degeneracy in Riemannian manifolds
zbMATH Open document ID: 0962.53009
published in: Calculus of Variations and Partial Differential Equations
publication date: 2001-06-18T00:00:00Z
review text: After some informal history and some Teichmüller and harmonic mapping theory of closed surfaces, following the exposition of \textit{A. Tromba} [``Teichmüller Theory in Riemannian Geometry'', Birkhäuser, Basel (1992; Zbl 0785.53001)], the author consider maps from a compact Riemann surface of genus \(\geq 2\) to a compact Riemannian manifold \((N,G)\) of strictly negative curvature. In that situation (essential uniqueness of harmonic representatives), the energy functional can be adjusted to give an energy \(\widetilde E\) whose critical points are the conformal harmonic maps. In a homotopy class \(\Sigma\) whose elements induce isomorphisms of fundamental groups, \(\widetilde E\) is proper -- so classical Morse theory applies. Much of this paper is devoted to generic non-degeneracy with respect to the metric \(G\) on \(N\), based on [\textit{A. Tromba}, J. Functional Anal. 23, 362-368 (1976; Zbl 0341.58005)]. Here the homomorphisms of the fundamental groups induced by the maps of \(\Sigma\) are assumed injective.
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 58B20
Mathematics Subject Classification ID: 53C43
zbMATH DE Number: 1454749
zbMATH Keywords: Plateau problem
zbMATH Keywords: conformal harmonic maps
zbMATH Keywords: Morse theory
Wikidata QID: Q115387755
author: Anthony J. Tromba
reviewed by: J. Eells
MaRDI profile type: Publication
full work available at URL: s005260050148
OpenAlex ID: W2024458308
DOI: 10.1007/S005260050148
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description: scientific article; zbMATH DE number 1454749