Search results for "22E25"
showing 9 items of 9 documents
Ahlfors-regular distances on the Heisenberg group without biLipschitz pieces
2015
We show that the Heisenberg group is not minimal in looking down. This answers Problem 11.15 in `Fractured fractals and broken dreams' by David and Semmes, or equivalently, Question 22 and hence also Question 24 in `Thirty-three yes or no questions about mappings, measures, and metrics' by Heinonen and Semmes. The non-minimality of the Heisenberg group is shown by giving an example of an Ahlfors $4$-regular metric space $X$ having big pieces of itself such that no Lipschitz map from a subset of $X$ to the Heisenberg group has image with positive measure, and by providing a Lipschitz map from the Heisenberg group to the space $X$ having as image the whole $X$. As part of proving the above re…
Isometries of nilpotent metric groups
2016
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 equipped with Carnot-Carath\'eodory distances. We study the regularity of isometries, i.e., distance-preserving homeomorphisms. Our first result is the analyticity of such maps between metric Lie groups. The second result is that if two metric Lie groups are connected and nilpotent then every isometry between the groups is the composition of a left translation and an isomorphism.…
Nowhere differentiable intrinsic Lipschitz graphs
2021
We construct intrinsic Lipschitz graphs in Carnot groups with the property that, at every point, there exist infinitely many different blow-up limits, none of which is a homogeneous subgroup. This provides counterexamples to a Rademacher theorem for intrinsic Lipschitz graphs.
Regularity properties of spheres in homogeneous groups
2015
We study left-invariant distances on Lie groups for which there exists a one-parameter family of homothetic automorphisms. The main examples are Carnot groups, in particular the Heisenberg group with the standard dilations. We are interested in criteria implying that, locally and away from the diagonal, the distance is Euclidean Lipschitz and, consequently, that the metric spheres are boundaries of Lipschitz domains in the Euclidean sense. In the first part of the paper, we consider geodesic distances. In this case, we actually prove the regularity of the distance in the more general context of sub-Finsler manifolds with no abnormal geodesics. Secondly, for general groups we identify an alg…
Metric equivalences of Heintze groups and applications to classifications in low dimension
2021
We approach the quasi-isometric classification questions on Lie groups by considering low dimensional cases and isometries alongside quasi-isometries. First, we present some new results related to quasi-isometries between Heintze groups. Then we will see how these results together with the existing tools related to isometries can be applied to groups of dimension 4 and 5 in particular. Thus we take steps towards determining all the equivalence classes of groups up to isometry and quasi-isometry. We completely solve the classification up to isometry for simply connected solvable groups in dimension 4, and for the subclass of groups of polynomial growth in dimension 5.
Sets with constant normal in Carnot groups: properties and examples
2019
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 constant-normal set in a Carnot group its sub-Riemannian-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 cone property. Such a cone will be a semisubgroup with nonempty interior that is canonically associated with the normal directio…
A Cornucopia of Carnot groups in Low Dimensions
2022
Abstract Stratified groups are those simply connected Lie groups whose Lie algebras admit a derivation for which the eigenspace with eigenvalue 1 is Lie generating. When a stratified group is equipped with a left-invariant path distance that is homogeneous with respect to the automorphisms induced by the derivation, this metric space is known as Carnot group. Carnot groups appear in several mathematical contexts. To understand their algebraic structure, it is useful to study some examples explicitly. In this work, we provide a list of low-dimensional stratified groups, express their Lie product, and present a basis of left-invariant vector fields, together with their respective left-invaria…
A Primer on Carnot Groups: Homogenous Groups, Carnot-Carathéodory Spaces, and Regularity of Their Isometries
2017
AbstractCarnot 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 remarks.We consider them as special cases of graded groups and as homogeneous metric spaces.We discuss the regularity of isometries in the general case of Carnot-Carathéodory spaces and of nilpotent metric Lie groups.
Sard property for the endpoint map on some Carnot groups
2016
In Carnot-Caratheodory or sub-Riemannian geometry, one of the major open problems is whether the conclusions of Sard's theorem holds for the endpoint map, a canonical map from an infinite-dimensional path space to the underlying finite-dimensional manifold. The set of critical values for the endpoint map is also known as abnormal set, being the set of endpoints of abnormal extremals leaving the base point. We prove that a strong version of Sard's property holds for all step-2 Carnot groups and several other classes of Lie groups endowed with left-invariant distributions. Namely, we prove that the abnormal set lies in a proper analytic subvariety. In doing so we examine several characterizat…