Grant 0330/2017 "Volumes of hyperbolic and Einstein manifolds"

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)
  • Clement Radu Popescu (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.

    Activity

    Publications
  • Cezar Joita, Mihai Tibar, Images of analytic map germs, preprint 2018.
  • Daniel Cibotaru, Sergiu Moroianu, Odd Pfaffian forms, preprint 2018.
  • H. Gaussier, Cezar Joita, On Runge neighborhoods of closures of domains biholomorphic to a ball, Geometric function theory in higher dimension, 63-66, Springer INdAM Ser., 26 (2017).
  • Cezar Joita, Mihai Tibar, Bifurcation set of multi-parameter families of complex curves, Journal of Topology 11, no. 3 (2018), 739-751.
  • Andrei Moroianu, Sergiu Moroianu, Liviu Ornea, Locally conformally Kahler manifolds with holomorphic Lee field, Differential Geom. Appl. 60, 33-38 (2018).

  • Conferences organized
  • S Moroianu, Topology and Geometry: A conference in memory of Stefan Papadima (1953-2018), Bucuresti, 28-31 mai 2018.

  • Talks at conferences
  • C Joita: "Bifurcation set of multi-parameter families of complex curves", The singular side, Université de Lille, May 16 - 17, 2018.
  • S Moroianu, Lectures on Hyperbolic Geometry, UAIC Iasi, oct. 2018.
  • S Moroianu, Geometry Day - 2017, UAIC Iasi, sept. 19, 2017.

  • Invited talks
  • S Moroianu, Université de Lorraine, Nancy, November 5, 2018.

    Scientific visits
  • C Joita: 07.05.2018 - 25.05.2018, Université de Lille
  • C Joita: 10.09.2018 - 21.09.2018, Université de Lille
  • S Moroianu, UAIC Iasi, 15-19 oct. 2018.
  • S Moroianu, Université de Lorraine, Metz, Nov. 5-9, 2018.

  • Participation in conferences/summer schools
  • C Anghel, Master classe "Surfaces", Marseille, May 28 - June 1, 2018.
  • R Stan, Master classe "Surfaces", Marseille, May 28 - June 1, 2018.
  • C Anghel, 6th Heidelberg Laureate Forum, Heidelberg, May 22-28, 2018.
  • R Stan, 6th Heidelberg Laureate Forum, Heidelberg, May 22-28, 2018.


  • Updated: November 12, 2018