## 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*