arXiv subjects

Séverine Rigot

Publications and source records attributed to Séverine Rigot.

6 recordsLinked to original sources

arXiv2022

Monotone sets and local minimizers for the perimeter in Carnot groups

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 sets played an important role in several works related to geometric measure theory issues in the Heisenberg setting. In this paper, we work in an arbitrary Carnot group and show that its monotone subsets are sets with locally finite perimeter that are local minimizers for the perimeter. Under an additional condition on the ambient Carnot group, we prove that their measure-theoretic interior and support are precisely monotone. We also prove topological and measure-theoretic properties of local minimizers for the perimeter whose interest is independent from the study of monotone sets. As a combination of our results, we get in particular a sufficient condition under which any monotone set admits measure-theoretic representatives that are precisely monotone.

math.MG↗

arXiv2022

Precisely monotone sets in step-2 rank-3 Carnot algebras

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-convex set with h-convex complement. Such sets have been introduced and classified in the Heisenberg setting by Cheeger and Kleiner in the 2010's. In the present paper, we study precisely monotone sets in the wider setting of step-2 Carnot groups, equivalently step-2 Carnot algebras. In addition to general properties, we prove a classification in step-2 rank-3 Carnot algebras that generalizes the classification already known in the Heisenberg setting using sublevel sets of h-affine functions. A significant novelty is that such sublevel sets can be different from half-spaces.

math.MG↗

arXiv2021

Horizontally affine functions on step-2 Carnot algebras

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 rank-$n$ Carnot algebra is isomorphic to the exterior algebra of $\mathbb{R}^n$. Using that every Carnot algebra can be written as a quotient of a free Carnot algebra, we shall deduce from the free case a description of h-affine functions on arbitrary step-2 Carnot algebras, together with several characterizations of those step-2 Carnot algebras where h-affine functions are affine in the usual sense of vector spaces. Our interest for h-affine functions stems from their relationship with a class of sets called precisely monotone, recently introduced in the literature, as well as from their relationship with minimal hypersurfaces.

math.MG↗

arXiv2020

Semmes surfaces and intrinsic Lipschitz graphs in the Heisenberg group

A Semmes surface in the Heisenberg group is a closed set $S$ that is upper Ahlfors-regular with codimension one and satisfies the following condition, referred to as Condition B. Every ball $B(x,r)$ with $x \in S$ and $0 < r < \operatorname{diam} S$ contains two balls with radii comparable to $r$ which are contained in different connected components of the complement of $S$. Analogous sets in Euclidean spaces were introduced by Semmes in the late $80$'s. We prove that Semmes surfaces in the Heisenberg group are lower Ahlfors-regular with codimension one and have big pieces of intrinsic Lipschitz graphs. In particular, our result applies to the boundary of chord-arc domains and of reduced isoperimetric sets.

math.CA↗

arXiv2019

Quantitative notions of rectifiability in the Heisenberg groups

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 the bilateral weak geometric lemma (BWGL), and show that sets with BPiLG satisfy the BWGL.

math.MG↗

arXiv2018

The Besicovitch covering property in the Heisenberg group revisited

The Besicovitch covering property (BCP) is known to be one of the fundamental tools in measure theory, and more generally, a usefull property for numerous purposes in analysis and geometry. We prove both sufficient and necessary criteria for the validity of BCP in the first Heisenberg group equipped with a homogeneous distance. Beyond recovering all previously known results about the validity or non validity of BCP in this setting, we get simple descriptions of new large classes of homogeneous distances satisfying BCP. We also obtain a full characterization of rotationally invariant distances for which BCP holds in the first Heisenberg group under mild regularity assumptions about their unit sphere.

math.MG↗