Solenoidal manifolds
type: Item
instance of: scholarly article
zbMATH Open document ID: 1323.57018
publication date: 2014-09-08T00:00:00Z
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 32F10
Mathematics Subject Classification ID: 32G15
zbMATH DE Number: 6341192
zbMATH Keywords: lamination
zbMATH Keywords: Riemann surface
zbMATH Keywords: uniformization
zbMATH Keywords: Teichmüller space
MaRDI profile type: Publication
full work available at URL: jsing.2014.9o
OpenAlex ID: W4253235206
DOI: 10.5427/JSING.2014.9O
author: Dennis Sullivan
title: Solenoidal manifolds
published in: Journal of Singularities
review text: In the paper under review, the author discusses the geometry of solenoidal surfaces (or more generally, laminar surfaces) and corresponding Teichmüller spaces. A \(k\)-dimensional laminar manifold is a compact Hausdorff space which is locally homeomorphic to a \(k\)-disk times a compact totally disconnected space. A solenoidal manifold is a laminar manifold which is locally homeomorphic to a disk times the Cantor set. We can speak of the structures on a laminar manifold, such as smooth, Riemannian, complex etc.NEWLINENEWLINE\noindent It is shown, rather sketchily, that any oriented solenoidal \(1\)-manifolds is the boundary of an oriented solenoidal surface (Theorem \(1\)). As for laminar (or solenoidal) surfaces, several theorems mostly with sketches of the proof are stated. Theorem \(3\) states \textit{A. Candel}'s result [Ann. Sci. Éc. Norm. Supér. (4) 26, No. 4, 489--516 (1993; Zbl 0785.57009)]: for any Riemannian laminar surface, the universal cover of every leaf is conformally the disk, or there is a nontrivial transverse invariant measure (both can happen). Also due to Candel (1993) and \textit{A. Verjovsky} [Contemp. Math. 58, 233--253 (1987; Zbl 0619.32017)], if every leaf of a Riemannian laminar surface is conformally covered by the disk, then the unique hyperbolic metric on each leaf is transversally continuous (Theorem \(4\)). The author's result [in: Topological methods in modern mathematics. Proceedings of a symposium in honor of John Milnor's sixtieth birthday, held at the State University of New York at Stony Brook, USA, June 14-June 21, 1991. Houston, TX: Publish or Perish, Inc.. 543--564 (1993; Zbl 0803.58018)] states that the space of hyperbolic structures on a laminar surface up to isometries isotopic to the identity (``Teichmüller space'') has the structure of a separable complex Banach manifold. The metric is the natural Teichmüller metric based on the minimal conformal distortion of a map, and the isotopy classes of homeomorphisms preserving a…
reviewed by: Shigeaki Miyoshi
description: scientific article; zbMATH DE number 6341192