Séance Séminaire

Séminaire Gaston Darboux

Filling Riemann Surfaces by Hyperbolic Schottky Manifolds of Negative Renormalized Volume

Given a hyperbolizable 3-manifold N, the renormalized volume is a real-valued function on the space of convex co-compact hyperbolic structures on the interior of N, which always have infinite hyperbolic volume. The simplest examples of convex co-compact hyperbolic 3-manifolds are the handlebodies, and, given a connected Riemann surface X of genus at least 2, we call Schottky filling of X a handlebody with boundary at infinity X. A question attributed to Maldacena asks whether given a connected Riemann surface X of genus at least two, there exists a Schottky filling of X of negative renormalized volume. We present an upper bound for the renormalized volume in terms of the genus and the hyperbolic curve lengths of a suitable pants decomposition of X, which allows us to positively answer the question of Maldacena for certain classes of Riemann surfaces.