## Grant 0330/2017 "Volumul varietatilor hiperbolice si Einstein"

Identifier: PN-III-P4-ID-PCE-2016-0330

Period: 2017-2019.

Financed by: ANCS-UEFISCDI contract 127/2017

Budget: 850.000 lei

Grant director: Sergiu
Moroianu

Team members:
Cezar Joita (Experienced Researcher)
Daniel Matei (Experienced Researcher)
Cipriana Anghel (Master student)
Rares Stan (Master student)

Abstract.
The current project aims to study, among other invariants of hyperbolic manifolds, the so-called renormalized volume of geometrically finite 3-manifolds, and in higher dimensions that of asymptotically hyperbolic Einstein manifolds. The renormalized volume depends on a choice of metric in the given conformal class on the ideal boundary. If we choose this metric to be the unique hyperbolic metric (the Fuchsian uniformization) we get a well-defined functional on the Teichmuller space of the ideal boundary. There are two separate problems we want to attack: proving positivity properties of the renormalized volume, and proving that the renormalized volume extends to the boundary of Teichmuller space as a Kahler potential for the Weil-Petersson metric. The first result in this direction was due to Krasnov-Schlenker, who showed that the Hessian of the renormalized volume functional at the Fuchsian locus equals the Weil-Petersson inner product, hence it is positive definite. In a joint paper with C. Ciobotaru, the project leader showed that the renormalized volume is positive on the open set of almost-fuchsian manifolds. The Kahler potential property was proved for compact Σ and arbitrary geometrically finite X without cusps of rank 1 by Guillarmou and the project leader. We are also interested in the geometry and topology of compact hyperbolic manifolds, with a focus on volumes, analytic and Reidemeister torsions, and cohomological invariants. We propose here an in-depth analysis of the relationship between the volume as a geometric invariant, and twisted cohomology as a topological one.

*Modificat: 21 noiembrie 2017*