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.