Universal Liouville action as a renormalized volume and its gradient flow

Abstract

The universal Liouville action (also known as the Loewner energy for Jordan curves) is a Kähler potential on the Weil-Petersson universal Teichmüller space, which is identified with the family of Weil-Petersson quasicircles via conformal welding. Our main result shows that, under regularity assumptions, the universal Liouville action equals the renormalized volume of the hyperbolic $3$-manifold bounded by the two Epstein-Poincaré surfaces associated with the quasicircle. We also study the gradient descent flow of the universal Liouville action for the Weil-Petersson metric and show that the flow always converges to the origin (the circle). This provides a bound of the Weil-Petersson distance to the origin by the universal Liouville action.

49 pages, 1 figure. Revised according to referee reports. To appear in Duke Math. J