minimal surfaces in sub-Riemannian geometry

Geometry Analysis and Dynamics on sub-Riemannian manifolds

**9.30-10.30**Giovanna Citti*(Università di Bologna)*

Title:*Convergence of total variation flow on the Heisenberg group to a minimal graph*

Abstract: In this talk we prove the convergence of the total variation flow to a minimal graph in the Heinseberg group, obtained in collaboration with L. Capogna and M.Manfredini. We first prove uniform Gaussian estimates for the heat kernels associated with a Riemannian Metric, on the Heisenberg group. which converges to a subriemannian one, when a suitable parameter epsilon goes to 0. Using these estimates we establish interior Schauder estimates for the total variation flow which are stables in espilon and in time. As a consequence we obtain subriemannian minimal surfaces as t tends to \infty.**10.30-11.00**Coffee break**11.00-12.00**Manuel Ritoré*(Universidad de Granada)*

Title:*A flux formula for C^1 area-stationary surfaces in H^1*

Abstract: An area-stationary surface in H^1 is a critical point of the sub-Riemannian perimeter under any variation induced by a compactly supported vector field in H^1. We shall show that this condition implies a flux type formula and discuss the applications. In particular, we shall show that the regular part of the surface is foliated by horizontal geodesics, a result already obtained by Chang, Hwang and Yang for t-graphs satisfying weakly the minimal surface equation.

**2.00-3.00**Davide Vittone*(Università di Padova)*

Title:*Height estimate for minimal surfaces in Heisenberg groups*

Abstract: We prove a height-estimate (distance from the tangent hyperplane) for local minima of the perimeter in the sub-Riemannian Heisenberg group. The estimate is in terms of a power of the excess (L^2-mean oscillation of the normal) and its proof is based on a new coarea formula for rectifiable sets in the Heisenberg group.**3.10-4.10**Benoit Daniel*(Université de Lorraine)*

Title:*Minimal surfaces in the Heisenberg group*

Abstract: In this talk we will give an overview of some results on minimal surfaces in the 3-dimensional Heisenberg group endowed with a left-invariant Riemannian metric. More precisely, we will focus on the existence of a Weierstrass-type representation and the relation with harmonic maps, on the construction of catenoid-type examples (joint work with L. Hauswirth) and on a Bernstein-type theorem by I. Fernandez and P. Mira.**4.10-4.45**Coffee break**4.45-5.45**Valentino Magnani*(Università di Pisa)*

Title:*Higher codimensional sub-Riemannian measures*

Abstract: We present the intrinsic notion of surface area for higher codimensional submanifolds in a Carnot group. This geometric measure can be seen as the natural sub-Riemannian measure associated with the submanifold, due to its special relationship with the Hausdorff measure constructed by the sub-Riemannian distance of the group. Several open questions will be addressed.

Andrea Malchiodi *(SISSA, Italy)*