Related papers: Regularity properties of spheres in homogeneous gr…

Carnot groups are distinguished spaces that are rich of structure: they are those Lie groups equipped with a path distance that is invariant by left-translations of the group and admit automorphisms that are dilations with respect to the…

We consider 2-step free-Carnot groups equipped with sub-Finsler distances. We prove that the metric spheres are codimension-one rectifiable from the Euclidean viewpoint. The result is obtained by studying how the Lipschitz constant for the…

In the sub-Riemannian Heisenberg group equipped with its Carnot-Caratheodory metric and with a Haar measure, we consider isodiametric sets, i.e. sets maximizing the measure among all sets with a given diameter. In particular, given an…

We consider Lie groups equipped with arbitrary distances. We only assume that the distance is left-invariant and induces the manifold topology. For brevity, we call such object metric Lie groups. Apart from Riemannian Lie groups,…

We characterize convex isoperimetric sets in the Heisenberg group endowed with horizontal perimeter. We first prove Sobolev regularity for a certain class of vector fields in the plane with bounded variation, related to the curvature…

In the Engel group with its Carnot group structure we study subsets of locally finite subRiemannian perimeter and possessing constant subRiemannian normal. We prove the rectifiability of such sets: more precisely we show that, in some…

We study isometric embeddings of a Euclidean space or a Heisenberg group into a higher dimensional Heisenberg group, where both the source and target space are equipped with an arbitrary left-invariant homogeneous distance that is not…

This book explores geometries defined by left-invariant distance functions on Lie groups, with a particular focus on nilpotent groups and Carnot groups equipped with geodesic distances. Geodesic left-invariant metrics are either…

We consider left-invariant distances $d$ on a Lie group $G$ with the property that there exists a multiplicative one-parameter group of Lie automorphisms $(0, \infty)\rightarrow\mathtt{Aut}(G)$, $\lambda\mapsto\delta_\lambda$, so that $…

In this work, we prove three types of results with the strategy that, together, the author believes these should imply the local version of Hilbert's Fifth problem. In a separate development, we construct a nontrivial topology for rings of…

We prove that the boundary of an almost minimizer of the intrinsic perimeter in a plentiful group can be approximated by intrinsic Lipschitz graphs. Plentiful groups are Carnot groups of step~$2$ whose center of the Lie algebra is generated…

Our main result is a positive answer to the question whether one can find homogeneous distances on the Heisenberg groups that have the Besicovitch Covering Property (BCP). This property is well known to be one of the fundamental tools of…

The classical isodiametric inequality in the Euclidean space says that balls maximize the volume among all sets with a given diameter. We consider in this paper the case of Carnot groups. We prove that for any Carnot group equipped with a…

The paper is devoted to the large scale geometry of the Heisenberg group $\mathbb H$ equipped with left-invariant Riemannian distances. We prove that two such distances have bounded difference if and only if they are asymptotic, i.e., their…

We prove boundary H\"older and Lipschitz regularity for a class of degenerate elliptic, second order, inhomogeneous equations in non-divergence form structured on the left-invariant vector fields of the Heisenberg group. Our focus is on the…

We define and study the harmonic curves on domains in $\mathbb{R}^n$ into the first Heisenberg group $\mathbb{H}^1$. These are the $C^2$-regular mappings which are critical points of the second Dirichlet energy and satisfy the weak…

In this paper we prove that isoperimetric sets in three-dimensional homogeneous spaces diffeomorphic to $\mathbb{R}^3$ are topological balls. We also prove that in three-dimensional homogeneous spheres isopermetric sets are either…

We prove $L^p$-Hardy inequalities with distance to the boundary for domains in the Heisenberg group ${\mathbb{H}}^n$, $n\geq 1$. Our results are based on a certain geometric condition. This is first implemented for the Euclidean distance in…

We prove that if the shape of the metric unit ball in a homogeneous group enjoys a precise symmetry property, then the associated distance yields the standard form of the area formula. The result applies to some classes of smooth and…

Take a riemanniann nilmanifold, lift its metric on its universal cover. In that way one obtains a metric invariant under the action of some co-compact subgroup. We use it to define metric balls and then study the spectrum of the laplacian…