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Enrico Le Donne

Publications and source records attributed to Enrico Le Donne.

At least 19 recordsLinked to original sources
arXiv2026

Sard property for rank 2 polarizations in metabelian Lie groups

We provide sharp bounds on the dimension of the abnormal set for rank $2$ polarizations on metabelian Lie groups, establishing the Sard property for the end-point map of such groups. The proof is based on a novel approach that makes essential use of tools from tame geometry. We also obtain bounds for the dimension of the Goh-abnormal set for metabelian Lie groups where the codimension of the derived subgroup is at most $2$, with no assumption on the rank of the polarization. We thus infer that these polarized groups, equipped with sub-Riemannian structures, satisfy the minimizing Sard property.

math.DG↗
arXiv2026

Quantitative versions of Pansu Asymptotic Theorem and of Mitchell Tangent Theorem

We quantitatively study the speed of convergence of geodesic Lie groups to their metric limits. For nilpotent geodesic Lie groups, we give estimates on the difference of the original metrics and the asymptotic metrics, while for general geodesic Lie groups, we give similar estimates for the difference of the original metrics and the tangent metrics. In both settings, our results sharpen existing bounds in the literature.

math.DG↗
arXiv2025

Normal Curves in Sub-Finsler Lie Groups: Branching for Strongly Convex Norms and Face Stability for Polyhedral Norms

We consider Lie groups equipped with left-invariant subbundles of their tangent bundles and norms on them. On these sub-Finsler structures, we study the normal curves in the sense of control theory. We revisit the Pontryagin Maximum Principle using tools from convex analysis, expressing the normal equation as a differential inclusion involving the subdifferential of the dual norm. In addition to several properties of normal curves, we discuss their existence, the possibility of branching, and local optimality. Finally, we focus on polyhedral norms and show that normal curves have controls that locally take values in a single face of a sphere with respect to the norm.

math.DG↗
arXiv2025

Asymptotics of Riemannian Lie groups with nilpotency step 2

We derive sharp estimates comparing asymptotic Riemannian or sub-Riemannian metrics in 2-step nilpotent Lie groups. For each metric, we construct a Carnot metric whose square remains at bounded distance from the square of the original metric. In particular, we deduce the analogue of a conjectire by Burago-Margulis: every 2-step nilpotent Riemannian Lie group is at bounded distance from its asymptotic cone. As a consequence, we obtain a refined estimate of the error term in the asymptotic expansion of the volume of the (sub-)Riemannian metric balls. To achive this, we develop a novel technique to efficiently perturb rectifiable curves modifying their endpoints in a prescribed vertical direction.

math.DG↗
arXiv2025

Quasi-conformal VS quasi-isometric equivalence in spaces with controlled growth

We study conditions under which quasi-conformal homeomorphisms are quasi-isometries. We show that if two nilpotent geodesic Lie groups are quasi-conformally homeomorphic, then they are quasi-isometrically equivalent. We also give more general results beyond the nilpotent case. In particular, we show that quasi-conformal homeomorphisms between geodesic Lie groups are quasi-isometries whenever the spaces have strict parabolic or hyperbolic conformal type. As a consequence, quasi-conformal homeomorphisms between geodesic Lie groups with infinite fundamental group are quasi-isometries. The statements for Lie groups are deduced from a more general study on metric measure spaces with uniformly locally bounded geometry.

math.MG↗
arXiv2025

Hypergenerated Carnot groups

In this paper we provide an algebraic characterization of those stratified groups in which boundaries with locally constant normal are locally flat. We show that these groups, which we call hypergenerated, are exactly the stratified groups where embeddings of non-characteristic hypersurfaces are locally bi-Lipschitz. Finally, we extend these results to submanifolds of arbitrary codimension.

math.MG↗
arXiv2024

Carnot rectifiability and Alberti representations

A metric measure space is said to be Carnot-rectifiable if it can be covered up to a null set by countably many biLipschitz images of compact sets of a fixed Carnot group. In this paper, we give several characterisations of such notion of rectifiability both in terms of Alberti representations of the measure and in terms of differentiability of Lipschitz maps with values in Carnot groups. In order to obtain this characterisation, we develop and study the analogue of the notion of Lipschitz differentiability space by Cheeger, using Carnot groups and Pansu derivatives as models. We call such metric measure spaces Pansu differentiability spaces (PDS).

math.MG↗
arXiv2024

Metric Lie Groups. Carnot-Carathéodory spaces from the homogeneous viewpoint

This book explores geometries defined by left-invariant distance functions on Lie groups, with a particular focus on nilpotent groups and Carnot groups equipped with geodesic distances. Geodesic left-invariant metrics are either sub-Riemannian or their generalizations, known as sub-Finsler geometries or Carnot-Carathéodory metrics. The primary objective is to illustrate how these non-smooth geometries, together with a Lie group structure, manifest in various mathematical fields, including metric geometry and geometric group theory. Additionally, the book demonstrates the role of metric Lie groups, particularly Carnot groups, in the following contexts: (a) as asymptotic cones of nilpotent groups; (b) as parabolic boundaries of rank-one symmetric spaces and, more broadly, of homogeneous negatively curved Riemannian manifolds; (c) as limits of Riemannian manifolds and tangents of sub-Riemannian manifolds.

math.DG↗
arXiv2024

Metabelian distributions and sub-Riemannian geodesics

We begin by characterizing metabelian distributions in terms of principal bundle structures. Then, we prove that in sub-Riemannian manifolds with metabelian distributions of rank $r$, the projection of strictly singular trajectories to some $r$-dimensional manifold must remain within an analytic variety. As a consequence, for rank-2 metabelian distributions, geodesics are of class $C^1$.

math.DG↗
arXiv2023

A rectifiability result for finite-perimeter sets in Carnot groups

In the setting of Carnot groups, we are concerned with the rectifiability problem for subsets that have finite sub-Riemannian perimeter. We introduce a new notion of rectifiability that is, possibly, weaker than the one introduced by Franchi, Serapioni, and Serra Cassano. Namely, we consider subsets $Γ$ that, similarly to intrinsic Lipschitz graphs, have a cone property: there exists an open dilation-invariant subset $C$ whose translations by elements in $Γ$ don't intersect $Γ$. However, a priori the cone $C$ may not have any horizontal directions in its interior. In every Carnot group, we prove that the reduced boundary of every finite-perimeter subset can be covered by countably many subsets that have such a cone property. The cones are related to the semigroups generated by the horizontal half-spaces determined by the normal directions. We further study the case when one can find horizontal directions in the interior of the cones, in which case we infer that finite-perimeter subsets are countably rectifiable with respect to intrinsic Lipschitz graphs. A sufficient condition for this to hold is the existence of a horizontal one-parameter subgroup that is not an abnormal curve. As an application, we verify that this property holds in every filiform group, of either first or second kind.

math.AP↗
arXiv2023

Intrinsically Lipschitz sections and applications to metric groups

We introduce a notion of intrinsically Lipschitz graphs in the context of metric spaces. This is a broad generalization of what in Carnot groups has been considered by Franchi, Serapioni, and Serra Cassano, and later by many others. We proceed by focusing our attention on the graphs as subsets of a metric space given by the image of a section of a quotient map and we require an intrinsically Lipschitz condition. We shall not have any function on a topological product, not we shall consider a metric on the base of the quotient map. Our results are: an Ascoli-Arzelà compactness theorem, an Ahlfors regularity theorem, and some extension theorems for partially defined intrinsically Lipschitz sections. Known results by Franchi, Serapioni, and Serra Cassano, and by Vittone will be our corollaries.

math.MG↗
arXiv2023

Sympletic reduction of the sub-Riemannian geodesic flow for metabelian nilpotent groups

We consider nilpotent Lie groups for which the derived subgroup is abelian. We equip them with subRiemannian metrics and we study the normal Hamiltonian flow on the cotangent bundle. We show a correspondence between normal trajectories and polynomial Hamiltonians in some euclidean space. We use the aforementioned correspondence to give a criterion for the integrability of the normal Hamiltonian flow. As an immediate consequence, we show that in Engel-type groups the flow of the normal Hamiltonian is integrable. For Carnot groups that are semidirect products of two abelian groups, we give a set of conditions that normal trajectories must fulfill to be globally length-minimizing. Our results are based on a symplectic reduction procedure.

math.DG↗
arXiv2023

Notions of null sets in infinite-dimensional Carnot groups

We study several notions of null sets on infinite-dimensional Carnot groups. We prove that a set is Aronszajn null if and only if it is null with respect to measures that are convolutions of absolutely continuous (CAC) measures on Carnot subgroups. The CAC measures are the non-abelian analogue of cube measures. In the case of infinite-dimensional Heisenberg-like groups we also show that being null in the previous senses is equivalent to being null for all heat kernel measures. Additionally, we show that infinite-dimensional Carnot groups that have locally compact commutator subgroups have the structure of Banach manifolds. There are a number of open questions included as well.

math.MG↗
arXiv2023

Escape from compact sets of normal curves in Carnot groups

In the setting of subFinsler Carnot groups, we consider curves that satisfy the normal equation coming from the Pontryagin Maximum Principle. We show that, unless it is constant, each such a curve leaves every compact set, quantitatively. Namely, the distance between the points at time 0 and time $t$ grows at least of the order of $t^{1/s}$, where $s$ denotes the step of the Carnot group. In particular, in subFinsler Carnot groups there are no periodic normal geodesics.

math.DG↗
arXiv2022

Rough similarity of left-invariant Riemannian metrics on some Lie groups

We consider Lie groups that are either Heintze groups or Sol-type groups, which generalize the three-dimensional Lie group SOL. We prove that all left-invariant Riemannian metrics on each such a Lie group are roughly similar via the identity. This allows us to reformulate in a common framework former results by Le Donne-Xie, Eskin-Fisher-Whyte, Carrasco Piaggio, and recent results of Ferragut and Kleiner-Müller-Xie, on quasiisometries of these solvable groups.

math.GR↗
arXiv2022

Blowups and blowdowns of geodesics in Carnot groups

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 of Carnot geodesics are geodesics in some groups of lower nilpotency step. Namely, every blowup curve of every geodesic in every Carnot group is still a geodesic in the group modulo its last layer. Then as a consequence we get that in every sub-Riemannian manifold any $s$ times iterated tangent of any geodesic is a line, where $s$ is the step of the sub-Riemannian manifold in question. With a similar approach, we also show that blowdown curves of geodesics in sub-Riemannian Carnot groups are contained in subgroups of lower rank. This latter result is also extended to rough geodesics.

math.MG↗
arXiv2021

Nilpotent groups and biLipschitz embeddings into $L^1$

We prove that if a simply connected nilpotent Lie group quasi-isometrically embeds into an $L^1$ space, then it is abelian. We reach this conclusion by proving that every Carnot group that biLipschitz embeds into $L^1$ is abelian. Our proof follows the work of Cheeger and Kleiner, by considering the pull-back distance of a Lipschitz map into $L^1$ and representing it using a cut measure. We show that such cut measures, and the induced distances, can be blown up and the blown-up cut measure is supported on "generic" tangents of the original sets. By repeating such a blow-up procedure, one obtains a cut measure supported on half-spaces. This differentiation result then is used to prove that bi-Lipschitz embeddings can not exist in the non-abelian settings.

math.MG↗