Lipschitz regularity in some geometric problems
type: Item
instance of: scholarly article
title: Lipschitz regularity in some geometric problems
zbMATH Open document ID: 1073.53040
author: Abdelghani Zeghib
published in: Geometriae Dedicata
publication date: 2005-03-15T00:00:00Z
review text: The starting fact for the recent paper is the following: Let \(U\) be an open subset of \(\mathbb{R}^2\). Let \(\mathcal{F}\) be a family of disjoint (straight) line segments which are connected components of intersections \(l\cap U\) with \(l\) being a stright line. Then \(\mathcal{F}\) is locally Lipschitz, that is the map \(u:S\to S^1\) defined in the support \(S\subset U\) of \(\mathcal{F}\) which associates to a line segment its unit tangent vector is locally Lipschitz. Moreover, the Lipschitz constant (which is a function on \(U\)) depends only on \(U\) and not on \(\mathcal{F}\). The aim of the article is to analyse to what extent the above fact generalizes to codimension one foliations by 'rigid' leaves. In particular, the author answers the followig question: What is the required regularity for a homeomorphism of \(\mathbb{R}\) to be a holonomy map of a foliation of an open set of \(\mathbb{R}^2\) by graphs of polynomials with bounded degree.
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 53C29
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 2145824
zbMATH Keywords: Lipschitz
zbMATH Keywords: line
zbMATH Keywords: graph of a polynomial
zbMATH Keywords: codimension one foliation
reviewed by: Andrzej Piatkowski
MaRDI profile type: Publication
DOI: 10.1023/B:GEOM.0000049090.87058.3A
description: scientific article; zbMATH DE number 2145824