## 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)
Open positions for Master and PhD students

Open positions for a Master and a PhD 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: 08 august 2017*