The existence of polarized \(F\)-structures on volume collapsed 4- manifolds
type: Item
instance of: scholarly article
title: The existence of polarized \(F\)-structures on volume collapsed 4- manifolds
zbMATH Open document ID: 0813.53030
DOI: 10.1007/BF01896239
published in: Geometric and Functional Analysis. GAFA
publication date: 1993-12-20T00:00:00Z
full work available at URL: https://eudml.org/doc/58143
review text: An \(F\)-structure on a manifold \(M\) is a family of local torus actions on \(M\) satisfying certain consistency conditions on overlaps. An \(F\)- structure is said to be polarized if all isotropy groups of the local torus actions are of finite order. The minimal volume of \(M\) is the infimum of volumes of metrics whose sectional curvature satisfies \(| K|\leq 1\). \textit{J. Cheeger} and \textit{M. Gromov} [J. Differ. Geom. 23, 309-346 (1986; Zbl 0606.53028)] and \textit{K. Fukaya} [ibid. 28, No. 1, 1- 21 (1988; Zbl 0652.53031)] show that if \(M\) admits a polarized \(F\)- structure then its minimal volume is zero. The author proves that the minimal volume of a 4-manifold is either 0 or uniformly bounded away from 0, by finding a polarized \(F\)-structure whenever the minimal volume is sufficiently small.
reviewed by: Mikhail G. Katz
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 53C10
zbMATH DE Number: 465140
zbMATH Keywords: gap conjecture
zbMATH Keywords: \(F\)-structure
zbMATH Keywords: torus actions
zbMATH Keywords: minimal volume
author: Xiaochun Rong
MaRDI profile type: Publication
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OpenAlex ID: W2105545680
description: scientific article; zbMATH DE number 465140