Calculus of Variations and Geometric Measure Theory

S. Don - E. Le Donne - T. Moisala - D. Vittone

A rectifiability result for finite-perimeter sets in Carnot groups

created by ledonne on 03 Dec 2019
modified by vittone on 27 Sep 2023


Accepted Paper

Inserted: 3 dec 2019
Last Updated: 27 sep 2023

Journal: Indiana Univ. Math. J.
Year: 2019

ArXiv: 1912.00493 PDF


In the setting of Carnot groups, we are concerned with the rectifiability problem for subsets that have finite sub-Riemannian perimeter. We introduce a new notion of rectifiability that is possibly, weaker than the one introduced by Franchi, Serapioni, and Serra Cassano. Namely, we consider subsets $\Gamma$ that, similarly to intrinsic Lipschitz graphs, have a cone property: there exists an open dilation-invariant subset $C$ whose translations by elements in $\Gamma$ don't intersect $\Gamma$. However, a priori the cone $C$ may not have any horizontal directions in its interior. In every Carnot group, we prove that the reduced boundary of every finite-perimeter subset can be covered by countably many subsets that have such a cone property. The cones are related to the semigroups generated by the horizontal half-spaces determined by the normal directions. We further study the case when one can find horizontal directions in the interior of the cones, in which case we infer that finite-perimeter subsets are countably rectifiable with respect to intrinsic Lipschitz graphs. A sufficient condition for this to hold is the existence of a horizontal one-parameter subgroup that is not an abnormal curve. As an application, we verify that this property holds in every filiform group, of either first or second type.

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