## 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

*Reports*

Final report
Report 2019
Report 2018
Report 2017

*Publications*

Luis Renato G. Dias, Cezar Joita, Mihai Tibar, *
Atypical points at infinity and algorithmic detection of the bifurcation locus of real polynomials*,
Mathematische Zeitschrift **298** (2021), no. 3-4, 1545--1558.
Cezar Joita, Mihai Tibar, *
Images of analytic map germs and singular fibrations*,
European Journal of Mathematics **6** (2020), 888-904.
Cezar Joita, Mihai Tibar, *
The local image problem for complex analytic map germs*, Arkiv for Matematik,
**59** (2021), 345-358.
Daniel Cibotaru, Sergiu Moroianu, *
Odd Pfaffian forms*, Bulletin of the Brazilian Math. Society. **52** (2021), 915-976
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 was one of the organizers of the section
Algebraic, Complex and Differential Geometry and Topology,
of the Ninth Congress of Romanian Mathematicians,
Galati, June 28 - July 3, 2019.
C Joita was one of the organizers of the section
Real and Complex Analysis, Potential Theory,
of the Ninth Congress of Romanian Mathematicians,
Galati, June 28 - July 3, 2019.
R Popescu was one of the organizers of the section
Selected Topics in Complex and Differential Geometry, Topology, and Singularities,
of the Ninth Congress of Romanian Mathematicians,
Galati, June 28 - July 3, 2019.
C Joita and S Moroianu were among the organizers of the conference
Bucharest Conference on Geometry and Physics,
Bucharest, Sept. 2-6, 2019.
S Moroianu was one of the organizers of the conference
Workshop on Riemannian and Kähler Geometry,
IMAR Bucharest,
April 15-19, 2019.
S Moroianu was one of the organizers of the conference
Topology and Geometry:
A conference in memory of Stefan Papadima (1953-2018),
Bucuresti, 28-31 mai 2018.

*Talks at conferences*
S Moroianu, "Tetraedre hiperbolice ideale"
SSMR conference XXIII,
Pitesti, Oct. 11-13, 2019.
S Moroianu, ``The renormalized volume in hyperbolic geometry''
IMAR 70 conference,
Bucharest, Oct. 4--5, 2019.
S Moroianu, ``Gauss-Bonnet-Chern formulas on singular manifolds''
Conference of the Mathematical Society of the Republic of Moldova
dedicated to the 55th anniversary of the Institute of Mathematics and Computer Science,
Chisinau, Sept 28- Oct. 1, 2019.
R Popescu,
Topology, Geometry, and Dynamics: Rokhlin 100,
Saint Petersburg, Rusia, Aug. 19-23, 2019.
R Popescu,
The 14-th Int'l Workshop on Differential Geometry and its applications,
Petroleum-Gas University of Ploiesti (UPG), July 9 - 11, 2019.
C Joita,
15th Romanian-Finnish Analysis Seminar,
University of Turku, Finland, June 10-12, 2019.
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*

C Joita,
Coverings of 1-convex surfaces, IMAR Monthly Colloquium, Dec. 5, 2019.
D Matei,
Resonance varieties and matrix tree theorems, Topology Seminar, oct.
2018, University of Tokyo, Japan.
D Matei,
Volumes of knot and link cone-manifolds, Graduate Topology Seminar,
nov. 2018, University of Tokyo, Japan.
D Matei,
Real singularities and contact structures, Workshop on Topology of Singularities,
nov. 2018, Keio University, Tokyo, Japan.
S Moroianu,
Université de Lorraine, Nancy, November 5, 2018.

*Scientific visits*
C Joita: 10.24.2019 - 24.10.2019, Université de Lille
S Moroianu, CIRM Marseille, 5.05.2019 - 10.05.2010.
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 29, 2019*