Euclid meets Heisenberg

In this thesis, we present four original results that extend classical Euclidean counterparts to the sub-Riemannian setting. The thesis is organized in six chapters and one appendix. In Chapter 1 and 2 we introduce the main definitions and some preliminary results that are going to be used in the last four chapters, where the original contributions are presented. In Chapter 3 we prove that that the diameter of small balls in C^{1,1} sub-Riemannian manifolds equals twice the radius. We also prove that, when the regularity of the structure is further lowered to C^0, the diameter is arbitrarily close to twice the radius. Both results hold independently of Hörmander condition. In Chapter 4 we introduce the space SBV_X of special functions with bounded X-variation in Carnot-Carathéodory spaces and study its main properties. We also prove an approximation result, with respect to the BV_X topology, for SBV_X functions. In Chapter 5 we prove a Stepanov differentiability type theorem for intrinsic graphs in sub-Riemannian Heisenberg groups. In Chapter 6 we introduce and study the notion of C^1_H-regular submanifold with boundary in sub-Riemannian Heisenberg groups. As an application, we prove a version of Stokes' Theorem for C^1_H-regular submanifolds with boundary and for the Rumin complex of differential forms on Heisenberg groups. Finally, in Appendix A, we discuss some work in progress on the relationship between contact geometry and sub-Riemannian geometry.