Rectifiability in Carnot Groups

This thesis is devoted to the study of the theory of rectifiability of sets and measures in the non smooth context of Carnot groups. The focus is on the study of the notion of P-rectifiability and its relation with other notions of rectifiability in Carnot groups. A P-rectifiable measure of integer dimension h in a Carnot group is a Radon measure with positive lower and finite upper h-densities almost everywhere such that the tangent measures are almost everywhere Haar measures of homogeneous subgroups of the Carnot group of homogeneous dimension h. The results discussed in this thesis have been obtained in the papers [25, 31–33]. In Chapter 1 we shall revise the basic notions of Measure Theory, and we shall introduce Carnot groups with a special focus on the notions of rectifiability, intrinsic regular functions, and submanifolds. In Chapter 2 we summarize part of the results obtained in collaboration with A. Merlo in [31, 33]. We prove that in arbitrary Carnot groups PPrectifiable measures of dimension h with a unique complemented tangent almost everywhere have h-density. We also characterizeP-rectifiable measures with complemented tangents by means of a covering property with intrinsically differentiable graphs in Carnot groups. These results complement and extend in several directions the study by Mattila–Serapioni–Serra Cassano in the Heisenberg groups Hn. In Chapter 3 we give the proof of a Marstrand–Mattila type rectifiability criterion in Carnot groups for P-rectifiable measures with tangents that admit at least one normal complementary subgroup. This result extends to the Carnot setting the Marstrand–Mattila rectifiability criterion in Euclidean spaces. We exploit such a criterion to derive as a consequence the Preiss’s Theorem for one-dimensional Radon measures in the first Heisenberg group H1 endowed with the Korányi norm. The results in Chapter 3 have been obtained in collaboration with A. Merlo in [32]. In Chapter 4 we present the results obtained with E. Le Donne in [25]. In some Carnot group of homogeneous dimension 13 we construct an analytic hypersurface, which is also a C1H -hypersurface, that is purely unrectifiable with respect to Carnot groups of homogeneous dimension 12. This gives an example of a C1H-hypersurface that is not Pauls rectifiable. As a consequence Franchi–Serapioni–Serra Cassano’s notion of C1H -rectifiability differs from Pauls’s notion of rectifiability in arbitrary Carnot groups. We further present a proof of the fact that in Hn, with n ≥ 2, every Euclidean C∞-hypersurface can be almost everywhere covered by bi-Lipschitz images of subsets of codimension-one subgroups of Hn.