Combinatorial volume preserving flows and taut foliations
type: Item
instance of: scholarly article
title: Combinatorial volume preserving flows and taut foliations
zbMATH Open document ID: 0964.57021
author: David Gabai
published in: Commentarii Mathematici Helvetici
publication date: 2001-07-23T00:00:00Z
review text: The author proves a remarkably strong generalization of a theorem of \textit{R. Roussarie} [Publ. Math., Inst. Hautes Étud. Sci. 43 (1973), 101-141 (1974; Zbl 0356.57017)] and \textit{W. Thurston} [UC Berkeley, Ph. D. Thesis (1972)] to the situation of immersed, incompressible surfaces in closed, tautly foliated 3-manifolds of class \(C^0\). The original theorem concerned imbedded, incompressible surfaces in tautly foliated 3-manifolds of class \(C^2\). The author's result is that, even in the \(C^0\) case, the merely immersed surface is homotopic to an immersed surface that either lies entirely in a leaf or is transverse to the foliation except at isolated saddle tangencies. An elegant proof of the original Roussari-Thurston theorem is due to \textit{J. Hass} [Comment. Math. Helv. 61, 1-32 (1986; Zbl 0601.53024)]. He combines a theorem of \textit{D. Sullivan} [ibid. 54, 218-223 (1979; Zbl 0409.57025)] and a result of \textit{R. Schoen} and \textit{S.-T. Yau} [Ann. Math. (2) 110, 127-142 (1979; Zbl 0431.53051)]. The first of these theorems guarantees that the tautly foliated manifold admits a Riemannian metric in which each leaf is minimal. The second establishes that a \(\pi_1\)-injective map of a closed, orientable surface into a Riemannian 3-manifold is homotopic to a least area immersion. The author abstracts the ideas of minimal surface theory, Haken's normal surface theory and Thurston's proof of the Sullivan theorem to obtain an elementary and intuitive proof of the \(C^0\) result. An important tool is his notion of a combinatorial, volume-preserving flow.
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 1455390
zbMATH Keywords: incompressible surfaces
reviewed by: Lawrence Conlon
MaRDI profile type: Publication
full work available at URL: s000140050114
OpenAlex ID: W2067154656
Wikidata QID: Q124935552
DOI: 10.1007/S000140050114
description: scientific article; zbMATH DE number 1455390