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