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

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Matthieu Gendulphe

created by daniele on 30 Nov 2015

3 dec 2015
-- 16:00

Sala Seminari, DM, Pisa

Seminari di Geometria, Pisa

**Abstract.**

Let $X$ be a compact hyperbolic surface. The injectivity radius at a
point $p$ of $X$ is the radius of the largest embedded metric ball
centered at $p$, we denote it by $R_p(X)$.
The extrema of the injectivity radius have been widely studied using
different methods.
Schmutz and Bavard have developed a variational framework for the
study of $\min_p R_p(X)$ as a function over the Teichmüller space.
Bavard and Deblois have used some geometric decompositions to bound
$\max_p R_p(X)$ in terms of the topology of $X$.

In this talk I will present a variational approach for the study of
the injectivity radius, seen as a function over the Teichmüller space
of hyperbolic surfaces with a marked point.
I will show that this function is almost a Morse function, and I will
determine all its critical points. As a consequence I will obtain some
known inequalities due to Bavard and Deblois.