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THE RENORMALIZED VOLUME AND UNIFORMIZATION OF CONFORMAL STRUCTURES

Published online by Cambridge University Press:  30 June 2016

Colin Guillarmou
Affiliation:
DMA, U.M.R. 8553 CNRS, École Normale Superieure, 45 rue d’Ulm, 75230 Paris Cedex 05, France (cguillar@dma.ens.fr)
Sergiu Moroianu
Affiliation:
Institutul de Matematică al Academiei Române, P.O. Box 1-764, RO-014700 Bucharest, Romania (moroianu@alum.mit.edu)
Jean-Marc Schlenker
Affiliation:
University of Luxembourg, Campus Kirchberg, Mathematics Research Unit, BLG, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg, Luxembourg (jean-marc.schlenker@uni.lu)

Abstract

We study the renormalized volume of asymptotically hyperbolic Einstein (AHE in short) manifolds $(M,g)$ when the conformal boundary $\unicode[STIX]{x2202}M$ has dimension $n$ even. Its definition depends on the choice of metric $h_{0}$ on $\unicode[STIX]{x2202}M$ in the conformal class at infinity determined by $g$, we denote it by $\text{Vol}_{R}(M,g;h_{0})$. We show that $\text{Vol}_{R}(M,g;\cdot )$ is a functional admitting a ‘Polyakov type’ formula in the conformal class $[h_{0}]$ and we describe the critical points as solutions of some non-linear equation $v_{n}(h_{0})=\text{constant}$, satisfied in particular by Einstein metrics. When $n=2$, choosing extremizers in the conformal class amounts to uniformizing the surface, while if $n=4$ this amounts to solving the $\unicode[STIX]{x1D70E}_{2}$-Yamabe problem. Next, we consider the variation of $\text{Vol}_{R}(M,\cdot ;\cdot )$ along a curve of AHE metrics $g^{t}$ with boundary metric $h_{0}^{t}$ and we use this to show that, provided conformal classes can be (locally) parametrized by metrics $h$ solving $v_{n}(h)=\text{constant}$ and $\text{Vol}(\unicode[STIX]{x2202}M,h)=1$, the set of ends of AHE manifolds (up to diffeomorphisms isotopic to the identity) can be viewed as a Lagrangian submanifold in the cotangent space to the space ${\mathcal{T}}(\unicode[STIX]{x2202}M)$ of conformal structures on $\unicode[STIX]{x2202}M$. We obtain, as a consequence, a higher-dimensional version of McMullen’s quasi-Fuchsian reciprocity. We finally show that conformal classes admitting negatively curved Einstein metrics are local minima for the renormalized volume for a warped product type filling.

Type
Research Article
Copyright
© Cambridge University Press 2016 

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Footnotes

C. G. was partially supported by the A.N.R. project ACG ANR-10-BLAN-0105. S. M. was partially supported by the CNCS project PN-II-RU-TE-2011-3-0053; he thanks the Fondation des Sciences Mathématiques de Paris and the École Normale Supérieure for additional support. J.-M. S. was partially supported by the A.N.R. through projects ETTT, ANR-09-BLAN-0116-01, and ACG, ANR-10-BLAN-0105.

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