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Regularity problems in sub-Riemannian structures

PRIN 2022 F4F2LH - CUP J53D23003760006

Analysis in sub-Riemannian (SR) manifolds deals with totally degenerate structures which are not Riemannian at any point. A SR structure is a regular manifold with a sub-bundle of the tangent bundle, called horizontal space, and a metric on it. Local generators of this sub-bundle, called horizontal vector fields, encode the directions of propagation of mechanical, physical, biological or geometrical phenomena. All the relevant differential objects of a SR manifold are expressed in terms of these vector-fields. The degenerate and anisotropic character of the structure calls for new tools in the study of the regularity properties of geodesics, minimal surfaces, solutions to PDEs and to optimal transport problems.