Comparison between renormalized volume and area in the Weil--Petersson topology

Abstract

We prove that the renormalized volume between the two Epstein surfaces associated with a Weil--Petersson quasicircle is continuous with respect to the Weil--Petersson topology. As a consequence, we establish that the universal Liouville action is proportional to this renormalized volume for every Weil--Petersson quasicircle. We then establish an inequality relating this renormalized volume to the renormalized area introduced by Alexakis--Mazzeo. To this end, we compare both quantities with the Hilbert--Schmidt norm of the associated Grunsky operator, which can be expressed in terms of a nonlocal Lagrangian introduced by O'Hara--Solanes.

arXiv comment31 pages. v2: added a comparison between the renormalized area of minimal surfaces and renormalized volume between Epstein surfaces