Optimal transport and sub-Riemannian manifolds

Geometry Analysis and Dynamics on sub-Riemannian manifolds

**9:00-9:50**Séverine Rigot*(Université Nice Sophia Antipolis)*

Title:*Monge's transport problem for distance cost in metric spaces*

Abstract: In this talk we will address the problem of the existence of solutions to Monge's transport problem in metric spaces. We will present a strategy that has been successfully used in the case of the sub-Riemannian Heisenberg group (joint work with L. De Pascale). This strategy does not involve any Sudakov-type dimension reduction argument nor disintegration of measures and is by many aspects much more simplier. We will also discuss extension of this strategy to other metric spaces focusing on Carnot groups and more generally sub-Riemannian manifolds.**9:50-10:20**Coffee break**10:20-11:10**Luca Rizzi*(SISSA, Trieste)*

Title:*On Measure Contraction Properties in sub-Riemannian geometry*

Abstract: We discuss how the volume of measurable sets evolves upon geodesic contraction. A recent results by Juillet suggests that the behaviour is dramatically different when compared with the Riemannian one. For example, the volume of the metric ball of the Heisenberg group contracts to zero with an exponent equal to 5, whereas one would expect the topological (equal to 3) or at least the Hausdorff dimension (equal to 4).

We show that the same behaviour occurs for any sub-Riemannian manifold, and the critical exponent is a new dimensional invariant (the geodesic dimension).

A recent result by Rifford states that any (ideal) Carnot group satisfies the classical MCP(0,N) for some exponent N. It is unknown which is the optimal N. We conjecture that, at least for step 2 Carnot groups, the optimal exponent is the geodesic dimension. We prove this fact for the (3,6) Carnot group (which has geodesic dimension 14), and we discuss how the same techniques work for the general step 2, rank k, free Carnot group (whose geodesic dimension grows as k^3, while the topological and Hausdorff dimensions both grow as k^2).

This is a work in progress, in collaboration with D. Barilari.**11:20-12:10**Nicolas Juillet*(Université de Strasbourg)*

Title:*On the Brunn-Minkowski inequality*

Abstract: We will recall different versions of the Brunn-Minkowski inequality (interpolation of sets) and their connection to the isoperimetric problem and the theory of optimal transportation (interpolation of measures). We will examine the interpolation of sets on some sub-Riemannian manifolds, including the Heisenberg group and the Grushin plane.

**14:10-15:00**Roberta Ghezzi*(Université de Bourgogne)*

Title:*BV functions and sets of finite perimeter in sub-Riemannian manifolds*

Abstract: We give a notion of BV function on an oriented manifold where a volume form and a family of lower semicontinuous 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 Poincaré 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-Serra Cassano JGA 2003]. This is a joint work with L.Ambrosio and V.Magnani.**15:00-15:50**Martin Huesmann*(Universität Bonn)*

Title:*Optimal transport maps on non-branching metric measure spaces*

Abstract: Let (X, d, m) be a proper, non-branching, metric measure space. We show existence and uniqueness of optimal transport maps for cost written as non-decreasing and strictly convex functions of the distance, provided (X, d, m) satisfies a new weak property concerning the behavior of m under the shrinking of sets to points. This in particular covers spaces satisfying the measure contraction property. This is joint work with Fabio Cavalletti.**15:50-16:20**Coffee break**16:20-17:10**Manuel Ritoré*(Universidad de Granada)*

Title:*The Bernstein problem for C^1 intrinsic graph*

Abstract: We shall show that a locally area-minimizing complete intrinsic graph in H^1 is a vertical plane.

Alessio Figalli *(University of Texas at Austin)* and Ludovic Rifford *(Université Nice Sophia Antipolis)*