Carnot Groups Of Step Research Articles

A Bourgain–Brezis–Mironescu–Dávila theorem in Carnot groups of step two

A Bourgain–Brezis–Mironescu–Dávila theorem in Carnot groups of step two

A note on geometric algebras and control problems with SO(3)‐symmetries

We study the role of symmetries in control systems through the geometric algebra approach. We discuss two specific control problems on Carnot groups of step 2 invariant with respect to the action of . We understand the geodesics as the curves in suitable geometric algebras which allows us to assess a new algorithm for the local control.

Sets with constant normal in Carnot groups: properties and examples

We analyze subsets of Carnot groups that have intrinsic constant normal, as they appear in the blowup study of sets that have finite subRiemannian 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 constant-normal set in a Carnot group its subRiemannian-Lebesgue representative is regularly open, contractible, and its topological boundary coincides with the reduced boundary and with the measure-theoretic boundary. We infer these properties from a metric cone property. Such a cone will be a semisubgroup with nonempty interior that is canonically associated with the normal direction. We characterize the constant-normal sets exactly as those that are arbitrary unions of translations of such semisubgroups. Second, making use of such a characterization, we provide some pathological examples in the specific case of the free-Carnot group of step 3 and rank 2. Namely, we construct a constant normal set that, with respect to any Riemannian metric, is not of locally finite perimeter; we also construct an example with non-unique intrinsic blowup at some point, showing that it has different upper and lower subRiemannian density at the origin. Third, we show that in Carnot groups of step 4 or less, every constant-normal set is intrinsically rectifiable, in the sense of Franchi, Serapioni, and Serra Cassano.

Coadjoint Orbits and Time-Optimal Problems for Step-$$2$$ Free Nilpotent Lie Groups

The coadjoint orbits and the Casimir functions are described for free nilpotent Lie groups of step $$2$$ . The symplectic foliation consists of affine subspaces in the Lie coalgebra. Left-invariant time-optimal problems are considered on Carnot groups of step $$2$$ for which the set of admissible velocities is a strictly convex compact set in the first layer of the Lie algebra that contains the origin in its interior. The first integrals of the vertical subsystem of the Hamiltonian system of the Pontryagin maximum principle are described. For two-dimensional coadjoint orbits, the constancy and periodicity properties of the solutions of this subsystem, as well as the phase flow, are described.

Intrinsic Lipschitz graphs in Carnot groups of step 2

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.

Infinite Geodesics and Isometric Embeddings in Carnot Groups of Step 2

In the setting of step 2 sub-Finsler Carnot groups with strictly convex norms, we prove that all infinite geodesics are lines. It follows that for any other homogeneous distance, all geodesics are ...

Uniqueness of Viscosity Mean Curvature Flow Solution in Two Sub-Riemannian Structures

We provide a uniqueness result for a class of viscosity solutions to sub-Riemannian mean curvature flows. In a sub-Riemannian setting, uniqueness cannot be deduced by the comparison principle, which is known only for graphs and for radially symmetry surfaces. Here we use a definition of continuous viscosity solutions of sub-Riemannian mean curvature flows motivated from a regularized Riemannian approximation of the flow. With this definition, we prove that any continuous viscosity solution of the equation is a limit of a sequence of solutions of Riemannian flow and obtain as a consequence uniqueness and the comparison principle. The results are provided in the settings of both 3-dimensional rototranslation group ${SE}(2)$ and Carnot groups of step 2, which are particularly important due to their relation to the surface completion problem of a model of the visual cortex.

Universal differentiability sets and maximal directional derivatives in Carnot groups

We show that every Carnot group G of step 2 admits a Hausdorff dimension one ‘universal differentiability set’ N such that every Lipschitz map f:G→R is Pansu differentiable at some point of N. This relies on the fact that existence of a maximal directional derivative of f at a point x implies Pansu differentiability at the same point x. We show that such an implication holds in Carnot groups of step 2 but fails in the Engel group which has step 3.

Pliability, or the Whitney extension theorem for curves in Carnot groups

The Whitney extension theorem is a classical result in analysis giving a necessary and sufficient condition for a function defined on a closed set to be extendable to the whole space with a given class of regularity. It has been adapted to several settings, among which the one of Carnot groups. However, the target space has generally been assumed to be equal to R^d for some d $\ge$ 1. We focus here on the extendability problem for general ordered pairs (G\_1,G\_2) (with G\_2 non-Abelian). We analyze in particular the case G\_1 = R and characterize the groups G\_2 for which the Whitney extension property holds, in terms of a newly introduced notion that we call pliability. Pliability happens to be related to rigidity as defined by Bryant an Hsu. We exploit this relation in order to provide examples of non-pliable Carnot groups, that is, Carnot groups so that the Whitney extension property does not hold. We use geometric control theory results on the accessibility of control affine systems in order to test the pliability of a Carnot group. In particular, we recover some recent results by Le Donne, Speight and Zimmermann about Lusin approximation in Carnot groups of step 2 and Whitney extension in Heisenberg groups. We extend such results to all pliable Carnot groups, and we show that the latter may be of arbitrarily large step.

A variant of Gromov's problem on Hölder equivalence of Carnot groups

It is unknown if there exists a locally α-Hölder homeomorphism f:R3→H1 for any 12<α≤23, although the identity map R3→H1 is locally 12-Hölder. More generally, Gromov asked: Given k and a Carnot group G, for which α does there exist a locally α-Hölder homeomorphism f:Rk→G? Here, we equip a Carnot group G with the Carnot–Carathéodory metric. In 2014, Balogh, Hajłasz, and Wildrick considered a variant of this problem. These authors proved that if k>n, there does not exist an injective, (12+)-Hölder mapping f:Rk→Hn that is also locally Lipschitz as a mapping into R2n+1. For their proof, they use the fact that Hn is purely k-unrectifiable for k>n. In this paper, we will extend their result from the Heisenberg group to model filiform groups and Carnot groups of step at most three. We will now require that the Carnot group is purely k-unrectifiable. The main key to our proof will be showing that (12+)-Hölder maps f:Rk→G that are locally Lipschitz into Euclidean space, are weakly contact. Proving weak contactness in these two settings requires understanding the relationship between the algebraic and metric structures of the Carnot group. We will use coordinates of the first and second kind for Carnot groups.

Dimension estimates for Kakeya sets defined in an axiomatic setting

In this dissertation we define a generalization of Kakeya sets in certain metric spaces. Kakeya sets in Euclidean spaces are sets of zero Lebesgue measure containing a segment of length one in every direction. A famous conjecture, known as Kakeya conjecture, states that the Hausdorff dimension of any Kakeya set should equal the dimension of the space. It was proved only in the plane, whereas in higher dimensions both geometric and arithmetic combinatorial methods were used to obtain partial results. In the first part of the thesis we define generalized Kakeya sets in metric spaces satisfying certain axioms. These allow us to prove some lower bounds for the Hausdorff dimension of generalized Kakeya sets using two methods introduced in the Euclidean context by Bourgain and Wolff. With this abstract setup we can deal with many special cases in a unified way, recovering some known results and proving new ones. In the second part we present various applications. We recover some of the known estimates for the classical Kakeya and Nikodym sets and for curved Kakeya sets. Moreover, we prove lower bounds for the dimension of sets containing a segment in a line through every point of a hyperplane and of an (n-1)-rectifiable set. We then show dimension estimates for Furstenberg type sets (already known in the plane) and for the classical Kakeya sets with respect to a metric that is homogeneous under non-isotropic dilations and in which balls are rectangular boxes with sides parallel to the coordinate axis. Finally, we prove lower bounds for the classical bounded Kakeya sets and a natural modification of them in Carnot groups of step two whose second layer has dimension one, such as the Heisenberg group. On the other hand, if the dimension is bigger than one we show that we cannot use this approach.

On the lack of semiconcavity of the subRiemannian distance in a class of Carnot groups

We show by explicit estimates that the SubRiemannian distance in a Carnot group of step two is locally semiconcave away from the diagonal if and only if the group does not contain abnormal minimizing curves. Moreover, we prove that local semiconcavity fails to hold in the step-3 Engel group, even in the weaker “horizontal” sense.

Remarks about the Besicovitch Covering Property in Carnot groups of step 3 and higher

Remarks about the Besicovitch Covering Property in Carnot groups of step 3 and higher

A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincaré inequality

Let $$\mathbb M $$ be a smooth connected manifold endowed with a smooth measure $$\mu $$ and a smooth locally subelliptic diffusion operator $$L$$ satisfying $$L1=0$$ , and which is symmetric with respect to $$\mu $$ . We show that if $$L$$ satisfies, with a non negative curvature parameter, the generalized curvature inequality introduced by the first and third named authors in http://arxiv.org/abs/1101.3590 , then the following properties hold: The key ingredient is the study of dimension dependent reverse log-Sobolev inequalities for the heat semigroup and corresponding non-linear reverse Harnack type inequalities. Our results apply in particular to all Sasakian manifolds whose horizontal Webster–Tanaka–Ricci curvature is nonnegative, all Carnot groups of step two, and to wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is nonnegative.

Double Ball Property for non-divergence horizontally elliptic operators on step two Carnot groups

Let \mathcal L be a linear second order horizontally elliptic operator on a Carnot group of step two. We assume \mathcal L in non-divergence form and with measurable coefficients. Then, we prove the Double Ball Property for the nonnegative sub-solutions of \mathcal L . With our result, in order to solve the Harnack inequality problem for this kind of operators, it becomes sufficient to prove the so called \epsilon - Critical Density .

Differential forms in Carnot groups: a Γ-convergence approach

Carnot groups (connected simply connected nilpotent stratified Lie groups) can be endowed with a complex (E0*, dc) of “intrinsic” differential forms. In this paper we prove that, in a free Carnot group of step κ, intrinsic 1-forms as well as their intrinsic differentials dc appear naturally as limits of usual “Riemannian” differentials de, e > 0. More precisely, we show that L2-energies associated with e−κde on 1 forms Γ-converge, as e → 0, to the energy associated with dc.

Lifting of convex functions on Carnot groups and lack of convexity for a gauge function

Let \({\mathbb{G}}\) be a Carnot group of step r and m generators and homogeneous dimension Q. Let \({\mathbb{F}_{m,r}}\) denote the free Lie group of step r and m generators. Let also \({\pi:\mathbb{F}_{m,r}\to\mathbb{G}}\) be a lifting map. We show that any horizontally convex function u on \({\mathbb{G}}\) lifts to a horizontally convex function \({u\circ \pi}\) on \({\mathbb{F}_{m,r}}\) (with respect to a suitable horizontal frame on \({\mathbb{F}_{m,r}}\)). One of the main aims of the paper is to exhibit an example of a sub-Laplacian \({\mathcal{L}=\sum_{j=1}^m X_j^2}\) on a Carnot group of step two such that the relevant \({\mathcal{L}}\)-gauge function d (i.e., d2-Q is the fundamental solution for \({\mathcal{L}}\)) is not h-convex with respect to the horizontal frame {X1, . . . , Xm}. This gives a negative answer to a question posed in Danielli et al. (Commun. Anal. Geom. 11 (2003), 263–341).

First-order regularity of convex functions on Carnot Groups

We prove that h-convex functions on Carnot groups of step two are locally Lipschitz continuous with respect to any intrinsic metric. We show that an additional measurability condition implies the local Lipschitz continuity of h-convex functions on arbitrary Carnot groups.

The sub-elliptic obstacle problem: [formula omitted] regularity of the free boundary in Carnot groups of step two

The sub-elliptic obstacle problem arises in various branches of the applied sciences, e.g., in mechanical engineering and robotics, mathematical finance, image reconstruction and neurophysiology. In the recent paper [Donatella Danielli, Nicola Garofalo, Sandro Salsa, Variational inequalities with lack of ellipticity. I. Optimal interior regularity and non-degeneracy of the free boundary, Indiana Univ. Math. J. 52 (2) (2003) 361–398; MR1976081 (2004c:35424)] it was proved that weak solutions to the sub-elliptic obstacle problem in a Carnot group belong to the Folland–Stein (optimal) Lipschitz class Γ loc 1 , 1 (the analogue of the well-known C loc 1 , 1 interior local regularity for the classical obstacle problem). However, the regularity of the free boundary remained a challenging open problem. In this paper we prove that, in Carnot groups of step r = 2 , the free boundary is (Euclidean) C 1 , α near points satisfying a certain thickness condition. This constitutes the sub-elliptic counterpart of a celebrated result due to Caffarelli [Luis A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math. 139 (3–4) (1977) 155–184; MR0454350 (56 #12601)].

Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of Hörmander type

We prove the hypoellipticity for systems of Hörmander type with constant coefficients in Carnot groups of step 2. This result is used to implement blow-up methods and prove partial regularity for local minimizers of non-convex functionals, and for solutions of non-linear systems which appear in the study of non-isotropic metric structures with scalings. We also establish estimates of the Hausdorff dimension of the singular set.