arXiv research

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

Study on mappings in Carnot groups, proving rigidity results.

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

Existence and rigidity results for lifts in Carnot groups.

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.

Rigidity theorem for flag manifolds in various dimensions.

Improved Sobolev mappings in Carnot groups with weaker assumptions.

In the present paper we study the rigidity of 2-step Carnot groups, or equivalently, of graded 2-step nilpotent Lie algebras. We prove the alternative that depending on bi-dimensions of the algebra, the Lie algebra structure makes it either always of infinite type or generically rigid, and we specify the bi-dimensions …

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

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

When a discrete group admits a convex-cocompact action on a non-compact rank-one symmetric space, there is a natural lower bound for the Hausdorff dimension of the limit set, given by the Ahlfors regular conformal dimension of the boundary of the group. We show that equality is achieved precisely when the group stabili…

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 (…

We show that the tangent cone at the identity is not a complete quasiconformal invariant for sub-Riemannian nilpotent groups. Namely, we show that there exists a nilpotent Lie group equipped with left invariant sub-Riemannian metric that is not locally quasiconformally equivalent to its tangent cone at the identity. In…

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…

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

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

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.

Commutes Pansu pullback with spectral complexes in Carnot groups.

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

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

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…

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.

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

Extends potential theory to Carnot groups, estimating Hausdorff dimension.

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…

We prove an optimal reverse Poincaré inequality for the heat semigroup generated by the sub-Laplacian on a Carnot group of any step. As an application we give new proofs of the isoperimetric inequality and of the boundedness of the Riesz transform in Carnot groups.

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…

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.

We develope basic geometric quantities and properties of hypersurfaces in Carnot groups.

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

We introduce a dynamical-systems approach for the study of the Sard problem in sub-Riemannian Carnot groups. We show that singular curves can be obtained by concatenating trajectories of suitable dynamical systems. As an applications, we positively answer the Sard problem in some classes of Carnot groups.

A Carnot group G is a connected, simply connected, nilpotent Lie group with stratified Lie algebra. Intrinsic regular surfaces in Carnot groups play the same role as C^1 surfaces in Euclidean spaces. As in Euclidean spaces, intrinsic regular surfaces can be locally defined in different ways: e.g. as non critical level …

Study spectral properties of sub-Laplacians in Carnot groups.

Study area and coarea formulas for graphs and submanifolds in Carnot groups.

Jet spaces on Carnot groups have a canonical Lie group structure.

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.