arXiv subjects
Publications and source records attributed to Valentino Magnani.
We find all intrinsic measures of $C^{1,1}$ smooth submanifolds in the Engel group, showing that they are equivalent to the corresponding $d$-dimensional spherical Hausdorff measure restricted to the submanifold. The integer $d$ is the degree of the submanifold. These results follow from a different approach to negligibility, based on a blow-up technique.
In this paper we establish the basic tools to develop the "Calculus" associated with group-valued continuously Pansu differentiable mappings. We develop the technical machinery on which all of our results rely. In particular, the linearization of addends appearing in the Baker-Campbell-Hausdorff formula is one of the major tools in several proofs. This allows us to characterize graded group-valued continuously Pansu differentiable mappings through a system of nonlinear first order PDEs, namely, the "contact equations". Through this approach we establish a quantitative estimate for the Pansu difference quotient. This key result is crucial to get the mean value inequality, the inverse mapping theorem, the rank theorem and the implicit function theorem for graded group-valued continuously Pansu differentiable mappings. All of these results represent the backbone to develop a reasonable "Calculus" in this context. On the other hand, these theorems rise the question of possible factorizations of stratified groups into inner semidirect products. In fact, we consider this algebraic issue to show that image-sets and level-sets of suitable continuously Pansu differentiable mappings are intrinsic graphs. These sets represent the intrinsic regular submanifolds modeled on a couple of fixed groups. Notice that the metric properties of these sets are completely different from the ones of classical submanifolds.
We obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measure. We explicitly compute this formula in some simple examples and we present a lower semicontinuity result for the spherical Hausdorff measure with respect to the weak convergence of currents. Another application is the proof of an intrinsic coarea formula for vector-valued mappings on the Heisenberg group.