Systoles and topological Morse functions for Riemann surfaces.
type: Item
instance of: scholarly article
title: Systoles and topological Morse functions for Riemann surfaces.
zbMATH Open document ID: 1033.32010
author: Paul Schmutz Schaller
published in: Journal of Differential Geometry
publication date: 2002-08-15T00:00:00Z
review text: Let \(M\) be a Riemann surface of genus \(g\) with \(n\) cusps, \(T(g,n)\) -- the Teichmueller space of \(M\) and \(\Gamma(g,n)\) be the corresponding mapping class group. The aim of this paper is the construction of topological Morse functions on \(T(g,n)\) which are invariant with respect to \(\Gamma(g,n)\) and proper on the moduli space \(T(g,n)/\Gamma(g,n).\) Until now, no such functions were known and the functions of this paper are the first examples. Let \(F = \{u_{1},\dots, u_{m}\}\) be a set of \(m\) marked, simple closed geodesics of \(M\) which fills up. For \(M \in T(g,n)\) let \(\text{Tan}(M)\) be the tangent space of \(T(g,n)\) in \(M.\) For \(\xi \in \text{Tan}(M)\) and \(u \in F\) let \(\xi(u) \in \mathbb R\) be the derivation, induced by \(\xi,\) of the length function \(L(u).\) Define the set of minima of \(F\) as \[ \begin{multlined} \text{Min}(F) = \{M \in T(g,n) \mid \text{for all } \xi \in \text{Tan}(M) \text{ either } \xi(u) = 0,\;\forall u \in F,\\ \text{or } \exists u, v \in F,\;\xi(u) > 0,\;\xi(v) < 0 \}.\end{multlined} \] For \(M \in T(g,n)\) define the vector space \[ W_{F}(M) = \{(\xi(u_{1}),\dots, \xi(u_{m})) \in \mathbb R^{m} \mid \xi \in \text{Tan}(M)\}. \] Then \(M \in \text{Min}(F)\) is called \(F\)-regular if \(\dim W_{F}\) is locally constant on \(\text{Min}(F)\) in \(M.\) Let now \[ \text{syst}: T(g,n) \rightarrow \mathbb R_{+} \] be the function which associates to \(M \in T(g,n)\) the length of a systole, a shortest simple closed geodesic; \(\text{syst}\) is continuous on \(T(g,n),\) but has plenty of corners. Let \(S(M)\) be the set of systoles of \(M.\) Theorem A. The function \(\text{syst}\) is a topological Morse function on \(T(g,n)\) if the following conditions hold: (i) for all \(M \in T(g,n)\) with \(M \in \text{Min}(F)\) (where \(F = S(M)),\) \(M\) is \(F\)-regular. (ii) for all \(M \in T(g,n)\) with \(M \in \partial \text{Min}(F)\) (where \(F = S(M)),\) \(M\) is strongly \(F\)-regular. If condition (ii) holds, then \(…
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 30F60
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 1782614
zbMATH Keywords: Riemann surface of genus \(g\) with \(n\) cusps
zbMATH Keywords: Teichmueller space
zbMATH Keywords: mapping class group
zbMATH Keywords: topological Morse functions
zbMATH Keywords: systole
zbMATH Keywords: shortest simple closed geodesic
reviewed by: V. V. Chueshev
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214425347
OpenAlex ID: W1581372407
Wikidata QID: Q115193697
DOI: 10.4310/JDG/1214425347
description: scientific article; zbMATH DE number 1782614