Fat Triangulations, Curvature and Quasiconformal Mappings

Emil Saucan,Meir Katchalski

doi:10.3390/axioms1020099

Abstract

Department of Mathematics and Computer Science, The Open University of Israel, 1 University Rd.,Raanana 43107, Israel* Author to whom correspondence should be addressed; E-Mail: semil@tx.technion.ac.il;Tel.: +972-4-8292896; Fax: +972-4-8294799.Received: 9 April 2012; in revised form: 31 May 2012 / Accepted: 11 June 2012 /Published: 4 July 2012Abstract: We investigate the interplay between the existence of fat triangulations,PLapproximations of Lipschitz–Killing curvatures and the existence of quasiconformalmappings. In particular we prove that if there exists a quasiconformal mapping betweentwo PLor smooth n-manifolds, then their Lipschitz–Killing curvatures are bilipschitzequivalent. An extension to the case of almost Riemannian manifolds, of a previous existenceresult of quasimeromorphic mappings on manifolds due to the ﬁrst author is also given.Keywords: fat triangulation; Lipschitz–Killing curvatures; quasimeromorphic mapping1. IntroductionThe present note is largely motivated by our theorem below, itself a continuation and generalizationof previous results of Martio and Srebro [1], Tukia [2] and Peltonen [3]:Theorem 1.1 ([4]) Let M

Highlights

IntroductionThe present note is largely motivated by our theorem below, itself a continuation and generalization of previous results of Martio and Srebro [1], Tukia [2] and Peltonen [3]: Theorem 1.1 ([4]) Let M n be a connected, oriented n-dimensional (n ≥ 2) submanifold of RN (for some N sufficiently large), with boundary, having a finite number of compact boundary components, and such that one of the following condition holds:
We have investigated in more detail this aspect of the role of curvature, and showed the possibility of constructing fat triangulations using solely intrinsic curvature, Ricci curvature, to be more precise, in [12,13] and, in a more general context, in [14]
The reverse direction, that is the role of fat triangulations in determining or approximating curvature(s) was shown in detail in [10]—see Theorem [10] below

Summary

Introduction

The present note is largely motivated by our theorem below, itself a continuation and generalization of previous results of Martio and Srebro [1], Tukia [2] and Peltonen [3]: Theorem 1.1 ([4]) Let M n be a connected, oriented n-dimensional (n ≥ 2) submanifold of RN (for some N sufficiently large), with boundary, having a finite number of compact boundary components, and such that one of the following condition holds:. Recall that quasiconformal mappings are defined as follows: Definition 1.2 Let (M, d), (N, ρ) be metric spaces and let f : (M, d) → (N, ρ) be a homeomorphism. Recall that fat triangulations ( called thick in some of the literature) are defined (in [10]) as follows: Definition 1.11 Let τ ⊂ Rn ; 0 ≤ k ≤ n be a k-dimensional simplex. The reverse direction, that is the role of fat triangulations in determining (in the P L case) or approximating (in the smooth case) curvature(s) was shown in detail in [10]—see Theorem [10] below It is this direction, and its connection with the existence of quasimeromorphic mappings, that we explore in this paper. Remark 1.22 In a sense, the theorem above can be considered, in view of the previous Remark, as the “reverse” of the result of [10], Section 8, regarding the convergence of the boundary measures

Quasiconformal Mappings Between Manifolds

Quasimeromorphic Mappings on Almost Riemmanian Manifolds

Final Remarks