On rectifiable measures in Carnot groups: representation.

This paper deals with the theory of rectifiability in arbitrary Carnot groups, and in particular with the study of the notion of P -rectifiable measure. First, we show that in arbitrary Carnot groups the natural infinitesimal definition of rectifiabile measure, i.e., the definition given in terms of the existence of flat tangent measures, is equivalent to the global definition given in terms of coverings with intrinsically differentiable graphs, i.e., graphs with flat Hausdorff tangents. In general we do not have the latter equivalence if we ask the covering to be made of intrinsically Lipschitz graphs. Second, we show a geometric area formula for the centered Hausdorff measure restricted to intrinsically differentiable graphs in arbitrary Carnot groups. The latter formula extends and strengthens other area formulae obtained in the literature in the context of Carnot groups. As an application, our analysis allows us to prove the intrinsic C 1 -rectifiability of almost all the preimages of a large class of Lipschitz functions between Carnot groups. In particular, from the latter result, we obtain that any geodesic sphere in a Carnot group equipped with an arbitrary left-invariant homogeneous distance is intrinsic C 1 -rectifiable.

On the constancy theorem for anisotropic energies through differential inclusions.

In this paper we study stationary graphs for functionals of geometric nature defined on currents or varifolds. The point of view we adopt is the one of differential inclusions, introduced in this context in the recent papers (De Lellis et al. in Geometric measure theory and differential inclusions, 2019. arXiv:1910.00335; Tione in Minimal graphs and differential inclusions. Commun Part Differ Equ 7:1-33, 2021). In particular, given a polyconvex integrand f, we define a set of matrices C f that allows us to rewrite the stationarity condition for a graph with multiplicity as a differential inclusion. Then we prove that if f is assumed to be non-negative, then in C f there is no T N ' configuration, thus recovering the main result of De Lellis et al. (Geometric measure theory and differential inclusions, 2019. arXiv:1910.00335) as a corollary. Finally, we show that if the hypothesis of non-negativity is dropped, one can not only find T N ' configurations in C f , but it is also possible to construct via convex integration a very degenerate stationary point with multiplicity.