Systoles and topological Morse functions for Riemann surfaces. (Q1609741)

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 \(M \in T(g,n)\) is a critical point of \(\text{syst}\) if and only if \(M \in \text{Min}(S(M)).\) Let now \(M \in T(g,n)\), \(n > 0.\) Let \(O\) be a cusp of \(M\) and let \(G{O}(g,n) \subset \Gamma(g,n)\) be the subgroup (of index \(n)\) of elements which fix \(O.\) Let \(R\) be a certain set of closed geodesic in \(M.\) For \(M \in T(g,n),\) let \(\text{syst}_R(M)\) be the length of a shortest element of \(R\) in \(M.\) Theorem B. On \(T(g,n), n > 0,\) \(\text{syst}_R(M)\) is a topological Morse function, invariant with respect to \(G{O}(g,n).\) Theorem E. For every \(g \geq 2\) there exists \(M_{g} \in T(g,0)\) which is a non-degenerate critical point of \(\text{syst}\) of index \(4g - 5.\)