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.
We consider submanifolds of sub-Riemannian Carnot groups with intrinsic $$C^1$$
C
1
regularity ($$C^1_H$$
C
H
1
). Our first main result is an area formula for $$C^1_H$$
C
H
1
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 $$C^1_H$$
C
H
1
submanifolds into level sets of a $$C^1_H$$
C
H
1
function.
We study left-invariant distances on Lie groups for which there exists a one-parameter family of homothetic automorphisms. The main examples are Carnot groups, in particular the Heisenberg group with the standard dilations. We are interested in criteria implying that, locally and away from the diagonal, the distance is Euclidean Lipschitz and, consequently, that the metric spheres are boundaries of Lipschitz domains in the Euclidean sense. In the first part of the paper, we consider geodesic distances. In this case, we actually prove the regularity of the distance in the more general context of sub-Finsler manifolds with no abnormal geodesics. Secondly, for general groups we identify an algebraic criterium in terms of the dilating automorphisms, which for example makes us conclude the regularity of homogeneous distances on the Heisenberg group. In such a group, we analyze in more details the geometry of metric spheres. We also provide examples of homogeneous groups where spheres presents cusps. 14 3.4. The sub-Finsler distance is Lipschitz in absence of singular geodesics 15 4. Regularity of spheres in graded groups 16
We construct intrinsic Lipschitz graphs in Carnot groups with the property that, at every point, there exist infinitely many different blow-up limits, none of which is a homogeneous subgroup. This provides counterexamples to a Rademacher theorem for intrinsic Lipschitz graphs.
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