On Renormalized Volume

Franco Vargas Pallete (Berkeley)

Renormalized volume is a quantity that gives a notion of volume for hyperbolic $3$-manifolds which have infi nite volume under the classical defi nition. Due to Krasnov and Schlenker, it is motivated by the AdS/CFT correspondence of string theory, and has been used for instance by Brock and Bromberg to bound Weil-Petersson translation distance of pseudo-Anosov maps by a factor of the volume of their hyperbolic mapping tori. In this talk I'll discuss its definition and basic properties, as well as my own work on local convexity and convergence under suitable limits for convex co-compact hyperbolic $3$-manifolds. These results give a detailed description of the critical points and infima of Renormalized Volume (while fixing topological type).