W-volume for planar domains with circular boundary

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.

16 pages, 1 figure. Comments are welcome!

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