Universal Liouville action as a renormalized volume and its gradient flow

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.