On the codimension of the abnormal set in step two Carnot groups

In this article we prove that the codimension of the abnormal set of the endpoint map for certain classes of Carnot groups of step 2 is at least three. Our result applies to all step 2 Carnot groups of dimension up to 7 and is a generalisation of a previous analogous result for step 2 free nilpotent groups

Fine properties of functions with bounded variation in Carnot-Carath\'eodory spaces

We study properties of functions with bounded variation in Carnot-Ca\-ra\-th\'eo\-do\-ry spaces. We prove their almost everywhere approximate differentiability and we examine their approximate discontinuity set and the decomposition of their distributional derivatives. Under an additional assumption on the space, called property $\mathcal R$, we show that almost all approximate discontinuities are of jump type and we study a representation formula for the jump part of the derivative

Regolarità delle geodetiche nei gruppi di Carnot

La tesi è dedicata allo studio della regolarità delle geodetiche in una particolare classe di varietà subriemanniane, i gruppi di Carnot. Sfruttando il formalismo Lagrangiano (si tratta di considerare funzionali dell'azione non continui e che possono assumere valori non finiti) enunciamo alcuni promettenti risultati parziali, che garantiscono la regolarità all'infuori di un chiuso di misura nulla

Height estimate and slicing formulas in the Heisenberg group

We prove a height-estimate (distance from the tangent hyperplane) for Lambda-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

A compactness result for BV functions in metric spaces

We prove a compactness result for bounded sequences (u_j) of functions with bounded variation in metric spaces (X,d_j) where the space X is fixed but the metric may vary with j. We also provide an application to Carnot\u2013Carath\ue9odory spaces

Some remarks about parametrizations of intrinsic regular surfaces in the Heisenberg group

We prove that, in general, H-regular surfaces in the Heisenberg group H1 are not bi-Lipschitz equivalent to the plane R2 en- dowed with the "parabolic" distance, which instead is the model space for C1 surfaces without characteristic points. In Heisenberg groups Hn, H-regular surfaces can be seen as intrinsic graphs: we show that such parametrizations do not belong to Sobolev classes of metric-space valued maps