Universal Liouville action as a renormalized volume and its gradient flow

2023/11/30 by Bridgeman, Martin, Bromberg, Kenneth, Pallete, Franco Vargas +1 · 2 citations
#Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2311.18767

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.

Cited by

• Volume functions and boundary data of 3-dimensional hyperbolic manifolds
• Onsager--Machlup functional for multiple SLEs

Related

• From the random geometry of conformally invariant systems to the Kähler geometry of universal Teichmüller space
• Liouville properties
• Liouville Theory and the Weil-Petersson Geometry of Moduli Space
• The Loewner Energy via the Renormalised Energy of Moving Frames
• The Weil-Petersson metric and the renormalized volume of hyperbolic 3-manifolds