Least area surfaces can have excess triple points
type: Item
instance of: scholarly article
title: Least area surfaces can have excess triple points
zbMATH Open document ID: 0636.53011
DOI: 10.1016/0040-9383(87)90006-1
author: Robert Gulliver
author: Peter Scott
published in: Topology
publication date: 1987-01-01T00:00:00Z
review text: It is known by results of \textit{R. Schoen} and \textit{S.-T. Yau} [Ann. Math., II. Ser. 110, 127-142 (1979; Zbl 0431.53051)] that in the homotopy class of any map f of a closed surface to a Riemannian 3-manifold which induces a monomorphism on fundamental groups there is an area minimizing immersion. By results of \textit{M. Freedman, J. Hass}, and \textit{P. Scott} [Invent. Math. 71, 609-642 (1983; Zbl 0482.53045)] such a least area map will minimize, over the homotopy class of f, a certain invariant D(f) which is related to the number, d(f), of double curves of f. In general D(f)\(\neq d(f)\) and examples show that d(f) need not be minimized by a least area map. However in the case of a torus \(D(f)=d(f).\) In the present paper an example is given of a least area map of a torus which has triple points but which is homotopic to a general position immersion without triple points. This shows that the proper measure of complexity in this subject is necessarily different from the conventional triple point, double curve count.
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 4036618
zbMATH Keywords: area minimizing immersion
zbMATH Keywords: least area map
zbMATH Keywords: double curves
zbMATH Keywords: torus
zbMATH Keywords: triple points
zbMATH Keywords: measure of complexity
MaRDI profile type: Publication
full work available at URL: 0040-9383(87)90006-1
OpenAlex ID: W2094884007
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description: scientific article; zbMATH DE number 4036618