THE RENORMALIZED VOLUME AND UNIFORMISATION OF CONFORMAL STRUCTURES

Colin Guillarmou (1) , Sergiu Moroianu (2) , Jean-Marc Schlenker (3)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 IMAR - "Simion Stoilow" Institute of Mathematics
3 IMT - Institut de Mathématiques de Toulouse UMR5219

We study the renormalized volume of asymptotically hyperbolic Einstein (AHE in short) manifolds (M, g) when the conformal boundary ∂M has dimension n even. Its definition depends on the choice of metric h0 on ∂M 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 σ2-Yamabe problem. Next, we consider the variation of VolR(M, ·; ·) along a curve of AHE metrics g t with boundary metric h t 0 and we use this to show that, provided conformal classes can be (locally) parametrized by metrics h solving vn(h) = ∂M vn(h)dvol h , 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 (∂M) of conformal structures on ∂M. We obtain as a consequence a higher-dimensional version of McMullen's quasifuchsian 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.

Domaines: Géométrie différentielle [math.DG]

Colin Guillarmou, Sergiu Moroianu, Jean-Marc Schlenker. THE RENORMALIZED VOLUME AND UNIFORMISATION OF CONFORMAL STRUCTURES. Journal of the Institute of Mathematics of Jussieu, 2018, 17 (4), pp.853--912. ⟨hal-00806380v2⟩