Valentino Magnani

An intrinsic measure for submanifolds in stratified groups

Abstract For each submanifold of a stratified group, we find a number and a measure only depending on its tangent bundle, the grading and the fixed Riemannian metric. In two step stratified groups, we show that such number and measure coincide with the Hausdorff dimension and with the spherical Hausdorff measure of the submanifold with respect to the Carnot-Carathéodory distance, respectively. Our main technical tool is an intrinsic blow-up at points of maximum degree. We also show that the intrinsic tangent cone to the submanifold at these points is always a subgroup. Finally, by direct computations in the Engel group, we show how our results can be extended to higher step stratified groups, provided the submanifold is sufficiently regular.

Blow-up of regular submanifolds in Heisenberg groups and applications

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.

Measure of submanifolds in the Engel group

We find all intrinsic measures of

NONEXISTENCE OF HORIZONTAL SOBOLEV SURFACES IN THE HEISENBERG GROUP

C^{1,1}

H-convex distributions in stratified groups

smooth submanifolds in the Engel group, showing that they are equivalent to the corresponding

Lipschitz continuity, Aleksandrov theorem and characterizations for H-convex functions

d

Characteristic points, rectifiability and perimeter measure on stratified groups

-dimensional spherical Hausdorff measure restricted to the submanifold. The integer

A Blow-up Theorem for regular hypersurfaces on nilpotent groups

d

On a general coarea inequality and applications.

is the degree of the submanifold. These results follow from a different approach to negligibility, based on a blow-up technique.

Non-horizontal submanifolds and coarea formula

Involutivity is a well known necessary condition for integrability of smooth tangent distributions. We show that this condition is still necessary for integrability with Sobolev surfaces. We specialize our study to the left invariant horizontal distribution of the first Heisenberg group ℍ 1 . Here we answer a question raised in a paper by Z.M. Balogh, R. Hoefer-Isenegger, and J.T. Tyson.