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This article is devoted to the variational study of two functions definedover some Teichmueller spaces of hyperbolic surfaces. One is the systole ofgeodesic loops based at some fixed point, and the other one is the systole ofarcs.\par For each of them we determine all the critical points. It appearsthat the systole of arcs is a topological Morse function, whereas the systoleof geodesic loops have some degenerate critical points. However, thesedegenerate critical points are in some sense the obvious one, and they do notinterfere in the variational study of the function.\par At a nondegeneratecritical point, the systolic curves (arcs or loops depending on the functioninvolved) decompose the surface into regular polygons. This enables a completeclassification of these points, and some explicit computations. In particularwe determine the global maxima of these functions. This generalizes optimalinequalities due to Bavard and Deblois. We also observe that there is only onelocal maximum, this was already proved in some cases by Deblois.\par Ourapproach is based on the geometric Vorono\''i theory developed by Bavard. Touse this variational framework, one has to show that the length functions (ofarcs or loops) have positive definite Hessians with respect to some system ofcoordinates for the Teichm\''uller space. Following a previous work, we chooseBonahon's shearing coordinates, and we compute explicitly the Hessian of thelength functions of geodesic loops. Then we use a characterization of thenondegenerate critical points due to Akrout.

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