W-volume for planar domains with circular boundary

TL;DR

This paper introduces the W-volume for planar domains with circular boundary, linking it to the Laplacian determinant and applying it to Schottky uniformization and Loewner energy, advancing understanding in conformal geometry and hyperbolic 3-manifolds.

Contribution

It extends Epstein maps to conformal metrics, defines the W-volume for domains with circular boundary, and connects it to Laplacian determinants and geometric invariants.

Findings

W-volume is a realization of the Laplacian determinant in hyperbolic space.

Bound on renormalized volume for Schottky uniformization of genus g surfaces.

Provides a hyperbolic realization of Loewner energy for Jordan curves.

Abstract

We extend the notion of Epstein maps to conformal metrics on submanifolds of the unit sphere $\mathbb{S}^n=\partial_\infty\mathbb{H}^{n+1}$. Using this construction for curves in $\mathbb{S}^2$, we define the W-volume for conformal metrics on domains in $\overline{\mathbb{C}}=\mathbb{S}^2$ with round circles as boundaries. We show that the W-volume is a realization in $\mathbb{H}^3$ of the determinant of the Laplacian. We use this and work of Osgood, Phillips and Sarnak to show that a classical Schottky uniformization of a genus g Riemann surface has renormalized volume bounded by $(6g-8)\pi$, and by $-2\pi$ under further assumptions. This gives a partial answer to a question of Maldacena. We also then provide a $\mathbb{H}^3$ realization of the Loewner energy of a $C^{2,\alpha}$ Jordan curve.