arXiv research

We study metric contraction properties for metric spaces associated with left-invariant sub-Riemannian metrics on Carnot groups. We show that ideal sub-Riemannian structures on Carnot groups satisfy such properties and give a lower bound of possible curvature exponents in terms of the datas.

The book explores Lie groups and Carnot-Carathéodory spaces, highlighting their applications in metric geometry and geometric group theory.

Undergrad project: Shows geodesics coincide in Heisenberg group under two metrics.

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 distance. We present the basic theory of Carnot groups together with several remar…

We examine the theory of metric currents of Ambrosio and Kirchheim in the setting of spaces admitting differentiable structures in the sense of Cheeger and Keith. We prove that metric forms which vanish in the sense of Cheeger on a set must also vanish when paired with currents concentrated along that set. From this we…

We focus our attention on the notion of intrinsic Lipschitz graphs, inside a special class of metric spaces i.e. the Carnot groups. More precisely, we provide a characterization of locally intrinsic Lipschitz functions in Carnot groups of step 2 in terms of their intrinsic distributional gradients.

Study maps in semidirect products of groups, proving Lipschitz properties without intrinsic dilations.

The study finds optimal curvature pinching in Heintze groups.

The paper studies metrics and geodesics on a quaternionic Heisenberg group.

Classifies metric lines in Engel-type groups, a step towards solving sub-Riemannian manifold problems.

New insights into integrability and rectifiability in sub-Riemannian geometry.

The paper develops techniques to study dynamical systems with Carnot metrics.

We prove that any corank 1 Carnot group of dimension $k+1$ equipped with a left-invariant measure satisfies the $\mathrm{MCP}(K,N)$ if and only if $K \leq 0$ and $N \geq k+3$. This generalizes the well known result by Juillet for the Heisenberg group $\mathbb{H}_{k+1}$ to a larger class of structures, which admit non-t…

We introduce a new class of unbounded model subdomains of $\mathbb{C}^2$ for the $\Box_b$ problem. Unlike previous finite type models, these domains need not be bounded by algebraic varieties. In this paper we obtain precise global estimates for the Carnot-Carathéodory metric induced on the boundary of such domains by …

We consider the rate of volume growth of large Carnot-Carathéodory metric balls on a class of unbounded model hypersurfaces in $\mathbb{C}^2$. When the hypersurface has a uniform global structure, we show that a metric ball of radius $δ\gg 1$ either has volume on the order of $δ^3$ or $δ^4$. We also give necessary and …

Study examines harmonic functions in sub-Riemannian and RCD settings.

In this paper we study the main geometric properties of the Carnot-Carathéodory (abbreviated CC) distance $\dc$ in the setting of $k$-step sub-Riemannian Carnot groups from many different points of view. An extensive study of the so-called normal CC-geodesics is given. We state and prove some related variational formul…

Sobolev mappings preserve the Rumin complex on contact manifolds.

This study classifies metric lines in jet space.

This paper provides some partial regularity results for geodesics (i.e., isometric images of intervals) in arbitrary sub-Riemannian and sub-Finsler manifolds. Our strategy is to study infinitesimal and asymptotic properties of geodesics in Carnot groups equipped with arbitrary sub-Finsler metrics. We show that tangents…

In this paper we study the invariant Carnot-Caratheodory metrics on $SU(2)\simeq S^3$, $SO(3)$ and $SL(2)$ induced by their Cartan decomposition and by the Killing form. Beside computing explicitly geodesics and conjugate loci, we compute the cut loci (globally) and we give the expression of the Carnot-Caratheodory dis…

Proves rectifiability for specific metric spaces with unique tangents.

Sharp estimates on 2-step nilpotent Lie groups' metrics and cones.

New definition of Rumin complex for nilpotent Lie groups.

The purpose of these notes is to explain parts of Gromov's survey of Carnot-Carathedory spaces, in the light of subsequent results of M. Rumin. Among the rich material provided by Gromov, most of which pertains to analysis on metric spaces, we choose to concentrate on the H{ö}lder equivalence problem for Carnot manifol…

We analyze subsets of Carnot groups that have intrinsic constant normal, as they appear in the blowup study of sets that have finite sub-Riemannian perimeter. The purpose of this paper is threefold. First, we prove some mild regularity and structural results in arbitrary Carnot groups. Namely, we show that for every co…

This paper is a sequel of arxiv:1709.09045 and deals with privileged coordinates and nilpotent approximation of Carnot manifolds. By a Carnot manifold it is meant a manifold equipped with a filtration by subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. In this paper, we single…

The study lists low-dimensional stratified groups and their properties.

We investigate the minimal surface problem in the three dimensional Heisenberg group, H, equipped with its standard Carnot-Caratheodory metric. Using a particular surface measure, we characterize minimal surfaces in terms of a sub-elliptic partial differential equation and prove an existence result for the Plateau prob…

This paper studies the infinitesimal structure of Carnot manifolds. By a Carnot manifold we mean a manifold together with a subbundle filtration of its tangent bundle which is compatible with the Lie bracket of vector fields. We introduce a notion of differential, called Carnot differential, for Carnot manifolds maps (…

Smooth $C^1$ contact maps are always smooth in rigid Carnot groups.

Gromov proposed to extract the (differential) geometric content of a sub-riemannian space exclusively from its Carnot-Carathéodory distance. One of the most striking features of a regular sub-riemannian space is that it has at any point a metric tangent space with the algebraic structure of a Carnot group, hence a homo…

New stability theorems for H-type Carnot groups established.

Investigates intrinsic Lipschitz sections in nonlinear quotient maps.

Improved Sobolev mappings in Carnot groups with weaker assumptions.

Curves in Carnot groups avoid compact sets, growing at least $t^{1/s}$.

New bounds on geodesic dimension and curvature exponent in Carnot groups.

We prove that H-type Carnot groups of rank $k$ and dimension $n$ satisfy the $\mathrm{MCP}(K,N)$ if and only if $K\leq 0$ and $N \geq k+3(n-k)$. The latter integer coincides with the geodesic dimension of the Carnot group. The same result holds true for the larger class of generalized H-type Carnot groups introduced in…

Carnot groups can be polarized if they have specific coordinate systems.

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, distinguished examples are sub-Riemannian Lie groups and, in particular, Carnot groups equ…

Compact currents and charges in Carnot groups proved.

ODE trajectories become abnormal curves in Carnot groups.

Study on mappings between nonrigid Carnot groups, proving quasisymmetric rigidity.

This paper studies rectifiability in Carnot groups and proves geometric area formulas.

Commutes Pansu pullback with spectral complexes in Carnot groups.

Study on mappings in Carnot groups, proving rigidity results.

The paper explores the Rumin complex and spectral sequence on Carnot groups.

This short note has been written as an Oberwolfach report for the workshop "Differentialgeometrie im Grossen". We discuss properties of metric spaces that at almost all points admit a tangent metric space. We explain why, under some mild assumptions, the tangents are almost surely subFinsler Carnot groups. We mention s…