Séverine Rigot
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View article: Monotone sets and local minimizers for the perimeter in Carnot groups
Monotone sets and local minimizers for the perimeter in Carnot groups Open
Monotone sets have been introduced about ten years ago by Cheeger and Kleiner who reduced the proof of the non biLipschitz embeddability of the Heisenberg group into $L^1$ to the classification of its monotone subsets. Later on, monotone s…
View article: Precisely monotone sets in step-2 rank-3 Carnot algebras
Precisely monotone sets in step-2 rank-3 Carnot algebras Open
A subset of a Carnot group is said to be precisely monotone if the restriction of its characteristic function to each integral curve of every left-invariant horizontal vector field is monotone. Equivalently, a precisely monotone set is a h…
View article: Horizontally affine functions on step-2 Carnot algebras
Horizontally affine functions on step-2 Carnot algebras Open
In this paper we introduce the notion of horizontally affine, h-affine in short, function and give a complete description of such functions on step-2 Carnot algebras. We show that the vector space of h-affine functions on the free step-2 r…
View article: Horizontally affine maps on step-two Carnot groups
Horizontally affine maps on step-two Carnot groups Open
In this paper we introduce the notion of horizontally affine, $h$-affine in short, maps on step-two Carnot groups. When the group is a free step-two Carnot group, we show that such class of maps has a rich structure related to the exterior…
View article: Semmes surfaces and intrinsic Lipschitz graphs in the Heisenberg group
Semmes surfaces and intrinsic Lipschitz graphs in the Heisenberg group Open
A Semmes surface in the Heisenberg group is a closed set that is upper Ahlfors-regular with codimension one and satisfies the following condition, referred to as Condition B. Every ball with and contains two balls with radii comparable…
View article: Quantitative notions of rectifiability in the Heisenberg groups
Quantitative notions of rectifiability in the Heisenberg groups Open
Several quantitative notions of rectifiability in the Heisenberg groups have emerged in the recent literature. In this paper we study the relationship between two of them, the big pieces of intrinsic Lipschitz graphs (BPiLG) condition and …
View article: Besicovitch Covering Property on graded groups and applications to measure differentiation
Besicovitch Covering Property on graded groups and applications to measure differentiation Open
We give a complete answer to which homogeneous groups admit homogeneous distances for which the Besicovitch Covering Property (BCP) holds. In particular, we prove that a stratified group admits homogeneous distances for which BCP holds if …
View article: S.Rigot - Besicovitch covering property in sub-Riemannian geometry
S.Rigot - Besicovitch covering property in sub-Riemannian geometry Open
The Besicovitch covering property originates from works of Besicovitch about differentiation of measures in Euclidean spaces. It can more generally be used as a usefull tool to deduce global properties of a metric space from local ones. We…
View article: Besicovitch covering property for homogeneous distances on the Heisenberg groups
Besicovitch covering property for homogeneous distances on the Heisenberg groups Open
We prove that the Besicovitch Covering Property (BCP) holds for homogeneous distances on the Heisenberg groups whose unit ball centered at the origin coincides with a Euclidean ball. We thus provide the first examples of homogeneous distan…
View article: Besicovitch Covering Property on graded groups and applications to measure differentiation
Besicovitch Covering Property on graded groups and applications to measure differentiation Open
We give a complete answer to which homogeneous groups admit homogeneous distances for which the Besicovitch Covering Property (BCP) holds. In particular, we prove that a stratified group admits homogeneous distances for which BCP holds if …