Convexity of the Renormalized volume of hyperbolic 3-manifolds
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CONVEXITY OF THE RENORMALIZED VOLUME OF HYPERBOLIC 3-MANIFOLDS By SERGIU MOROIANU Abstract. The Hessian of the renormalized volume of geometrically finite hyperbolic 3-manifolds without rank-1 cusps, computed at the hyperbolic metric ggeod with totally geodesic boundary of the convex core, is shown to be a strictly positive bilinear form on the tangent space to Teichmüller space. The metric ggeod is known from results of Bonahon and Storm to be an absolute minimum for the volume of the convex core. We deduce the strict convexity of the functional volume of the convex core at its minimum point. 1. Introduction. The renormalized volume is a functional on the moduli space of hyperbolic 3-manifolds of finite geometry. It has been introduced in this context by Krasnov [9], after initial work by Henningson and Skenderis [8] for more general Poincaré-Einstein manifolds. As 3-dimensional geometrically finite 3-manifolds are closely related to Riemann surfaces, VolR defines in a natural way a Kähler potential for the Weil-Petersson symplectic form on the Teichmüller space. This follows for quasi-fuchsian manifolds by the identity between the renormalized volume and the so-called classical Liouville action functional, a topological quantity known by work of Takhtadzhyan and Zograf [15] to provide a Kähler potential. For geometrically finite hyperbolic 3-manifolds without rank-1 cusps, the Kähler property of the renormalized volume was proved by Colin Guillarmou and the author in [6], by constructing a Chern-Simons theory on the Teichmüller space. The case of cusps of rank 1 is studied in a joint upcoming paper with Guillarmou and Frédéric Rochon. Here we look at a certain moduli space of complete, infinite-volume hyperbolic metrics g on a fixed 3-manifold X. The metrics we consider are geometrically finite quotients Γ\H3 (i.e., they admit a fundamental polyhedron with finitely many faces) and do not have cusps of rank 1, in the sense that every parabolic subgroup of Γ, if any, must have rank 2. We define the moduli space M as the quotient of the above set of metrics on X by the group Diff0 (X) of diffeomorphisms isotopic to the identity. The existence of such metrics on X implies that X is diffeomorphic to the interior of a manifold-with-boundary K. Let 2K be the smooth manifold obtained by doubling K across Σ. We make the following assumption throughout the paper: There exists on 2K a complete hyperbolic metric of finite volume. Manuscript received April 14, 2015; revised September 16, 2016. Research supported by the CNCS project PN-II-RU-TE-2012-3-0492. American Journal of Mathematics 139 (2017), 1379–1394. c  2017 by Johns Hopkins University Press. 1379 1380 S. MOROIANU It follows from Mostow-Prasad rigidity that up to a diffeomorphism of 2K isotopic to the identity, the boundary Σ = ∂K is totally geodesic for this metric. Since 2K must be aspherical and atoroidal, the connected components of Σ cannot be spheres or tori. Examples of manifolds where our assumption is not fulfilled are quasi-fuchsian manifolds and Schottky manifolds, since their double is not atoroidal. With the above assumption, a distinguished point ggeod in M is obtained from K by gluing infinite-volume funnels with vanishing Weingarten operator (see Section 2) to each boundary component of K. We call this metric the totally geodesic metric, and note that K is the convex core of (X,ggeod). It was remarked by Thurston, again as a simple consequence of Mostow rigidity, that ggeod is the unique metric in M with smooth boundary of the convex core. By work of Bonahon [2] it is known that the volume of the convex core Vol(C(X,g)) has a minimum at ggeod when viewed as a functional on M. When X is convex co-compact, i.e., without cusps, Storm [13] proved that the minimum point ggeod is strict. We shall apply here our results on VolR to deduce the convexity of Vol(C(X,g)) at this special point in M for X geometrically finite without cusps of rank 1, but possibly with cusps of rank 2 as in [2]. It is instructive to compare those...