Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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The injectivity radius of hyperbolic surfaces and some Morse functions over moduli spaces

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 Teichm├╝ller 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 Teichm├╝ller 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.

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