Upper bounds on Renormalized Volume for Schottky groups

Abstract

In this article we show that for any given Riemann surface $\hat{I}\pounds$ of genus $g$, we can bound (from above) the renormalized volume of a (hyperbolic) Schottky group with boundary at infinity conformal to $\hat{I}\pounds$ in terms of the genus and the combined extremal lengths on $\hat{I}\pounds$ of $(g-1)$ disjoint, non-homotopic, simple closed compressible curves. This result is used to partially answer a question posed by Maldacena about comparing renormalized volumes of Schottky and Fuchsian manifolds with the same conformal boundary.

17 pages, 5 figures. Typos and constants in the main theorem have been corrected. Proof of the main result was restructured. To appear in Mathematische Annalen