arXiv research

The H-type deviation measures how close step two Carnot groups are to H-type groups.

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

Study shows a specific Carnot group violates a curvature exponent bound.

Characterizes GM-groups via sub-Riemannian geometry properties.

New stability theorems for H-type Carnot groups established.

In this article we prove that the codimension of the abnormal set of the endpoint map for certain classes of Carnot groups of step 2 is at least three. Our result applies to all step 2 Carnot groups of dimension up to 7 and is a generalisation of a previous analogous result for step 2 free nilpotent groups.

Solves a long-standing problem on step-two groups with exact formulas.

A helical CR structure is a decomposition of a real Euclidean space into an even-dimensional horizontal subspace and its orthogonal vertical complement, together with an almost complex structure on the horizontal space and a marked vector in the vertical space. We prove an equivalence between such structures and step t…

The paper constructs optimal sub-Riemannian geodesics in specific Carnot groups.

Study exact formula and geodesics for Carnot-Carathéodory distance on 2-step groups.

We consider (smooth) solutions of the mean curvature flow of graphs over bounded domains in a Lie group free up to step two (and not necessarily nilpotent), endowed with a one parameter family of Riemannian metrics $σ_\e$ collapsing to a subRiemannian metric $σ_0$ as $\e\to 0$. We establish $C^{k,α}$ estimates for this…

In this note, we study the cut locus of the free, step two Carnot groups $\mathbb{G}_k$ with $k$ generators, equipped with their left-invariant Carnot-Carathéodory metric. In particular, we disprove the conjectures on the shape of the cut loci proposed in [Myasnichenko - 2002] and [Montanari, Morbidelli - 2016], by exh…

In this paper we study global Poincare inequalities on balls in a large class of sub-Riemannian manifolds satisfying the generalized curvature dimension inequality introduced by F.Baudoin and N.Garofalo. As a corollary, we prove the uniqueness of solutions for the subelliptic heat equation. Our results apply in particu…

By adapting some ideas of M. Ledoux \cite{ledoux2}, \cite{ledoux-stflour} and \cite{Led} to a sub-Riemannian framework we study Sobolev, Poincaré and isoperimetric inequalities associated to subelliptic diffusion operators that satisfy the generalized curvature dimension inequality that was introduced by F. Baudoin and…

In this paper, we consider a smooth connected finite-dimensional manifold $M$, an affine connection $\nabla$ with holonomy group $H^{\nabla}$ and $Δ$ a smooth completely non integrable distribution. We define the $Δ$-horizontal holonomy group $H^{\;\nabla}_Δ$ as the subgroup of $H^{\nabla}$ obtained by $\nabla$-paralle…

We study the topology of admissible-loop spaces on a step-two Carnot group G. We use a Morse-Bott theory argument to study the structure and the number of geodesics on G connecting the origin with a 'vertical' point (geodesics are critical points of the 'Energy' functional, defined on the loop space). These geodesics t…

Let $\M$ be a smooth connected non-compact manifold endowed with a smooth measure $μ$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $μ$. We show that if $L$ satisfies, with a non negative curvature parameter $ρ_1$, the generalized curvature inequality …

Let $\mathbb{M}$ be a smooth connected manifold endowed with a smooth measure $μ$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $μ$. We show that if $L$ satisfies, with a non negative curvature parameter, the generalized curvature inequality introduced…

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

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…

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 (…

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

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

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…

Improved Sobolev mappings in Carnot groups with weaker assumptions.

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

Compact currents and charges in Carnot groups proved.

ODE trajectories become abnormal curves in Carnot groups.

Maps in Carnot groups are equivalent to solutions of a PDE system.

Maps commuting with sub-Laplacians on Carnot groups are conformal.

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

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

We characterize the rigidity of Carnot groups in the class of $C^2$ contact maps in terms of complex characteristics. Furthermore, we obtain a Liouville type theorem for Carnot groups which states that 1-quasiconformal maps form finite dimensional Lie groups.

Commutes Pansu pullback with spectral complexes in Carnot groups.

Study on mappings in Carnot groups, proving rigidity results.

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

This paper characterizes semigenerated Carnot groups and applies it to rectifiability of perimeter sets.

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…

Existence and rigidity results for lifts in Carnot groups.

Study uniformly differentiable graphs in Carnot groups, proving area formulas.

Extends potential theory to Carnot groups, estimating Hausdorff dimension.

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.

Paper proves equivalence of derivatives for maps between Carnot groups.

In Carnot groups, directional pliability allows curve extensions and approximations.

Study Sard problem in step 2 and filiform Carnot groups.

Optimal quantization of measures on Carnot groups

A Carnot group $\mathbb{G}$ admits Lusin approximation for horizontal curves if for any absolutely continuous horizontal curve $γ$ in $\mathbb{G}$ and $\varepsilon>0$, there is a $C^1$ horizontal curve $Γ$ such that $Γ=γ$ and $Γ'=γ'$ outside a set of measure at most $\varepsilon$. We verify this property for free Carno…

The notion of curvature discussed in this paper is a far going generalization of the Riemannian sectional curvature. It was first introduced by Agrachev, Barilari and Rizzi in arXiv:1306.5318, and it is defined for a wide class of optimal control problems: a unified framework including geometric structures such as Riem…