An intrinsic measure for submanifolds in stratified groups

Towards Differential Calculus in stratified groups

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 di…

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 mapp…

Area of intrinsic graphs and coarea formula in Carnot Groups

We consider submanifolds of sub-Riemannian Carnot groups with intrinsic $C^1$ regularity ($C^1_H$). Our first main result is an area formula for $C^1_H$ intrinsic graphs; as an application, we deduce density properties for Hausdorff measures on rectifiable sets. Our second main result is a coarea formula for slicing $C^1_H$ submanifolds into level sets of a $C^1_H$ function.

亚黎曼海森堡群中带边界的子流形

Submanifolds with boundary in sub-Riemannian Heisenberg Groups

本文研究亚黎曼海森堡群中具有内在$C^1$正则性的带边界子流形，给出实例并建立涉及Rumin微分形式积分的斯托克斯定理。

Lipschitz functions on submanifolds in Heisenberg groups

We study the behavior of Lipschitz functions on intrinsic $C^1$ submanifolds of Heisenberg groups: our main result is their almost everywhere tangential Pansu differentiability. We also provide two applications: a Lusin-type approximation of Lipschitz functions on $\HH$-rectifiable sets, and a coarea formula on $\HH$-rectifiable sets that completes the program started in~\cite{JNGV}.

Regularity of sets with constant intrinsic normal in a class of Carnot groups

In this Note, we define a class of stratified Lie groups of arbitrary step (that are called ``groups of type $\star$'' throughout the paper), and we prove that, in these groups, sets with constant intrinsic normal are vertical halfspaces. As a consequence, the reduced boundary of a set of finite intrinsic perimeter in a group of type $\star$ is rectifiable in the intrinsic sense (De Giorgi's rectifiability theorem). This result extends the previous one proved by Franchi, Serapioni & Serra Cassano in step 2 groups.