The injectivity radius of hyperbolic surfaces and some Morse functions over moduli spaces

TL;DR

This paper studies the critical points of systole functions on hyperbolic surfaces within Teichmüller spaces, classifying them and identifying global maxima using geometric Voronoi theory and explicit Hessian computations.

Contribution

It provides a complete classification of critical points of systole functions, including explicit computations and the identification of global maxima, extending previous inequalities.

Findings

Systole of arcs is a topological Morse function.

Systole of geodesic loops has degenerate critical points.

There is only one local maximum for these functions.

Abstract

This article is devoted to the variational study of two functions defined over some Teichmueller spaces of hyperbolic surfaces. One is the systole of geodesic loops based at some fixed point, and the other one is the systole of arcs.\par For each of them we determine all the critical points. It appears that the systole of arcs is a topological Morse function, whereas the systole of geodesic loops have some degenerate critical points. However, these degenerate critical points are in some sense the obvious one, and they do not interfere in the variational study of the function.\par At a nondegenerate critical point, the systolic curves (arcs or loops depending on the function involved) decompose the surface into regular polygons. This enables a complete classification of these points, and some explicit computations. In particular we determine the global maxima of these functions. This generalizes optimal inequalities due to Bavard and Deblois. We also observe that there is only one local maximum, this was already proved in some cases by Deblois.\par Our approach is based on the geometric Vorono\''i theory developed by Bavard. To use this variational framework, one has to show that the length functions (of arcs or loops) have positive definite Hessians with respect to some system of coordinates for the Teichm\''uller space. Following a previous work, we choose Bonahon's shearing coordinates, and we compute explicitly the Hessian of the length functions of geodesic loops. Then we use a characterization of the nondegenerate critical points due to Akrout.