We study the H-regular surfaces, a class of intrinsic regular hypersurfaces in the setting of the Heisenberg group H^n = C^n x R = R^{2n+1} endowed with a left-invariant metric d equivalent to its Carnot-Carathéodory (CC) metric. Here hypersurface simply means topological codimension 1 surface and by the words "'intrinsic'" and "'regular" we mean, respectively, notions involving the group structure of H^n and its differential structure as CC manifold. In particular, we characterize these surfaces as intrinsic regular graphs inside H^n by studying the intrinsic regularity of the parameterizations and giving an area-type formula for their intrinsic surface measure
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.
We study the isoperimetric problem in Euclidean space endowed with a density. We first consider piecewise constant densities and examine particular cases related to the characteristic functions of half-planes, strips and balls. We also consider continuous modification of Gauss density in R 2 . Finally, we give a list of related open questions.
In the setting of the sub-Riemannian Heisenberg group H^n, we introduce and study the classes of t- and intrinsic graphs of bounded variation. For both notions we prove the existence of non-parametric area-minimizing surfaces, i.e., of graphs with the least possible area among those with the same boundary. For minimal graphs we also prove a local boundedness result which is sharp at least in the\ud
case of t-graphs in H^1
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.
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