# SÉMINAIRE DE GÉOMÉTRIE SOUS-RIEMANNIENNE

## Institut Henri Poincaré, Paris - 2012/13

The seminar takes place at the Institut Henri Poincaré, 11, rue Pierre et Marie Curie, Paris.

- Frequence: One/two seminars, once a month, on a wednesday afternoon.
- Topics: sub-Riemannian geometry and related fields.
- Organizers: Davide Barilari, Pierre Pansu.
- Some notes of the past seminars are available on this blog.

** The sub-Riemannian geometry seminar will resume next September for the 2013/14 session **

__--__

*PAST SEMINARS*

*Wednesday, May 22, 2013. (Salle 05)*

- 15.30 - 16.30.
**Bruno Franchi**(Universita di Bologna) -*Sharp a priori estimates for div-curl systems in Heisenberg groups*

*Abstract:*In this talk we present a result proved in collaboration with Annalisa Baldi. We prove a family of inequalities for differential forms in Heisenberg groups H1 and H2, that are the natural counterpart of a class of div-curl inequalities in de Rham's complex proved by Lanzani & Stein and Bourgain & Brezis.

- 16.45 - 17.45.
**Grégoire Charlot**(IF, Grenoble) -*On the heat kernel for Riemannian and sub-Riemannian structures*

*Abstract:*In a recent paper [1] the authors prove some results on the small time heat kernel asymptotics in Riemannian and sub-Riemannian cases, focusing on the case of pair of points (x,y) such that y is both cut and conjugate with respect to x. After a brief introduction, I will present some work in progress about the estimate of the small time heat kernel asymptotics for generic metrics, via the study of generic singularities of the exponential (sub)-Riemannian maps.

[1] D. Barilari, U. Boscain, R. Neel, Small time heat kernel asymptotics at the sub-Riemannian cut locus. JDG Vol 92, No.3, 2012, pp. 373-416.

*Wednesday, February 27, 2013. (Salle 05)*

- 15.30 - 16.30.
**Roberta Ghezzi**(SNS, Pisa) -*BV functions and finite perimeter sets in sub-Riemannian manifolds*

*Abstract:*We give a notion of BV function on an oriented manifold where a volume form and a family of l.s.c. quadratic forms are given. Using this notion, we generalize the structure theorem for BV functions that holds in the Euclidean case. When we consider sub-Riemannian manifolds, our definition coincide with the one given in the more general context of metric measure spaces which are doubling and support a PoincarĂ© inequality. We study finite perimeter sets of sub-Riemannian manifold, i.e., sets whose characteristic function is BV, and we prove a blowup theorem, generalizing the one obtained for step-2 Carnot groups in [Franchi-Serapioni-SerraCassano JGA 2003].

*Wednesday, January 16, 2013. (Salle 05)*

- 15.30 - 16.30.
**Michel Bonnefont**(Université Bordeaux 1) -*Measure doubling property and PoincarĂ© inequality in sub-elliptic geometry under a curvature-dimension criterion*

*Abstract:*In this seminar, I will present a curvature-dimension criterion in sub-elliptic geometry that generalizes the one of Bakry-Emery. I will present how one can obtain, starting from this criterion, gradient estimates of Li-Yau type and an inverse log-soboloev inequality for the semigroup. Finally, I will present how one can deduce the measure doubling property, gaussian estimates of the heat kernel and the PoincarĂ© inequality on balls. This is a joint work with F. Baudoin and N. Garofalo.

*Tuesday, December 4, 2012. (Salle 01)*

- 16.45 - 17.45.
**Nicolas Juillet**(IRMA, Strasbourg) -*Diffusion by optimal transport in the Heisenberg group*

*Abstract:*In this talk, we will present the subRiemannian Heisenberg group H and its hypoelliptic diffusion, the ``heat diffusion'' of H. We will show that in the Wasserstein space P_2(H), it is a curve driven by the gradient flow of the Boltzmann entropy Ent. Conversely any gradient flow curve of Ent satisfy the hypoelliptic heat equation (without any additional assumption).

*Wednesday, October 3, 2012. (Salle 05)*

- 15.30 - 16.30.
**Andrea Malchiodi**(SISSA, Trieste) -*A positive mass theorem for CR manifolds*

*Abstract:*We consider a class of CR manifold which we define as asymptotically Heisenberg, and for these we give a notion of mass, reminiscent of the ADM notion. From the solvability of the $\Box_b$ equation in a certain functional class, we prove positivity of the mass under the condition that the Webster curvature is positive and that the manifold is embeddable. We apply this result to the Yamabe problem for compact CR manifolds, assuming positivity of the Webster class and non-negativity of the CR Paneitz operator. - 16.45 - 17.45.
**Tsachik Gelander**(University of Jerusalem) -*On Benjamini-Schramm topology, Invariant Random Subgroups and asymptotic of L_2 invariants of locally symmetric spaces.*

More information:

- Organizers: Davide Barilari, Pierre Pansu.
- Current year: Séminaire de Géométrie sous-riemannienne - 2013/2014
- Previous year: Séminaire de Géométrie sous-riemannienne - 2011/2012