PL minimal surfaces in 3-manifolds
type: Item
instance of: scholarly article
title: PL minimal surfaces in 3-manifolds
zbMATH Open document ID: 0652.57005
author: William Jaco
author: J. Hyam Rubinstein
published in: Journal of Differential Geometry
publication date: 1988-01-01T00:00:00Z
review text: Starting with a fixed triangulation of a given 3-manifold M, the authors choose a metric on the 2-skeleton which is hyperbolic on every triangle in the sense of the hyperbolic plane. If a surface F in M meets the 2- skeleton of M transversally, then one can associate with F the total length of the intersection curves. The notion of a PL minimal surface F refers to a surface for which this length is stationary with respect to small variations of F. A mean curvature H can be introduced as the geodesic curvature of the intersection curves. In the paper the authors carry over results due to W. Meeks, P. Scott, S.-T. Yau and others to the case of PL minimal surfaces rather than ordinary minimal surfaces. In particular they give existence theorems of PL minimal surfaces in given normal homotopy classes, they discuss the exchange-and-roundoff-trick, and they give many results about the topology of M using PL minimal surfaces. It is impossible to mention all these in detail here. As an example, Corollary 2 says that a closed orientable irreducible 3-manifold with infinite fundamental group which is finitely covered by a Seifert fiber space must be a Seifert fiber space. Another theorem says that any covering of a \(P^ 2\)-irreducible 3-manifold is itself \(P^ 2\)-irreducible.
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 57Q35
zbMATH DE Number: 4064058
zbMATH Keywords: \(\pi_ 1\)-injective surface
zbMATH Keywords: triangulation of a 3-manifold
zbMATH Keywords: hyperbolic metric on the 2-skeleton
zbMATH Keywords: PL minimal surface
zbMATH Keywords: mean curvature
zbMATH Keywords: finitely covered by a Seifert fiber space
zbMATH Keywords: covering of a \(P^ 2\)-irreducible 3- manifold
Wikidata QID: Q115183788
reviewed by: Wolfgang Kühnel
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214442006
OpenAlex ID: W1574922261
DOI: 10.4310/JDG/1214442006
description: scientific article; zbMATH DE number 4064058