Abstract
AbstractWe 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.
Highlights
The interest towards Analysis and Geometry in Metric Spaces grew drastically in the last decades: a major effort has been devoted to the development of analytical tools for the study of geometric problems, and sub-Riemannian Geometry provided a fruitful setting for these investigations
The present paper aims at giving a contribution in this direction by providing some geometric integration formulae, namely: an area formula for submanifolds with C1 regularity, and a coarea formula for slicing such submanifolds into level sets of maps with C1 regularity
2.5 we prove that an Theorem holds for a submanifold ; namely, is an intrinsic graph, i.e., there exist complementary homogeneous subgroups W, V of G and a function φ : A → V defined on an open subset
Summary
The interest towards Analysis and Geometry in Metric Spaces grew drastically in the last decades: a major effort has been devoted to the development of analytical tools for the study of geometric problems, and sub-Riemannian Geometry provided a fruitful setting for these investigations. Theorem holds for a submanifold ; namely, is (locally) an intrinsic graph, i.e., there exist complementary homogeneous subgroups W, V of G and a function φ : A → V defined on an open subset. In Lemma 4.6 we prove Theorem 1.3 in the “linearized” case when both f and u are homogeneous group morphisms: in this case formula (4) holds with a constant coarea factor C(P, L) which depends only on the normal homogeneous subgroup P := ker f and on the homogeneous morphism L = u (on L| only). Riemannian coarea formula that is proved when the set is not a positive ψ Q-measure subset of G (i.e., in the notation of Theorem 1.3, when M = {0}). Theorem 1.7 (Coarea formula in Heisenberg groups) Consider an open set ⊂ Hn, a (Hn, Rm )-rectifiable set R ⊂.
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