The Renormalized Volume and Uniformization of Conformal Structures

We study the renormalized volume of asymptotically hyperbolic Einstein (AHE in short) manifolds $(M,g)$ when the conformal boundary [STIX]x2202M has dimension n even. Its definition depends on the choice of metric h₀ on [STIX]x2202M in the conformal class at infinity determined by g , we denote it by VolR(M,g;h₀) . We show that VolR(M,g;· ) is a functional admitting a ‘Polyakov type’ formula in the conformal class [h₀] and we describe the critical points as solutions of some non-linear equation vₙ(h₀)=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 [STIX]x1D70E₂ -Yamabe problem. Next, we consider the variation of VolR(M,· ;· ) along a curve of AHE metrics gᵗ with boundary metric h₀ᵗ and we use this to show that, provided conformal classes can be (locally) parametrized by metrics h solving vₙ(h)=constant and Vol([STIX]x2202M,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 T([STIX]x2202M) of conformal structures on [STIX]x2202M . 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.