Nowhere differentiable intrinsic Lipschitz graphs

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.

Intrinsic rectifiability via flat cones in the Heisenberg group

We give a geometric criterion for a topological surface in the first Heisenberg group to be an intrinsic Lipschitz graph, using planar cones instead of the usual open cones. peerReviewed

Sub-Finsler Horofunction Boundaries of the Heisenberg Group

We give a complete analytic and geometric description of the horofunction boundary for polygonal sub-Finsler metrics---that is, those that arise as asymptotic cones of word metrics---on the Heisenberg group. We develop theory for the more general case of horofunction boundaries in homogeneous groups by connecting horofunctions to Pansu derivatives of the distance function.

Regularity properties of spheres in homogeneous groups

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

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.

Lipschitz Functions on Submanifolds of Heisenberg Groups

Abstract We study the behavior of Lipschitz functions on intrinsic $C^1$ submanifolds of Heisenberg groups: our main result is their almost everywhere tangential Pansu differentiability. We also provide two applications: a Lusin-type approximation of Lipschitz functions on ${\mathbb {H}}$-rectifiable sets and a coarea formula on ${\mathbb {H}}$-rectifiable sets that completes the program started in [18].

Area-minimizing cones in the Heisenberg group H

We present a characterization of minimal cones of class \(C^2\) and \(C^1\) in the first Heisenberg group \(\mathbf{H}\), with an additional set of examples of minimal cones that are not of class \(C^1\).

Lipschitz Carnot-Carathéodory Structures and their Limits

AbstractIn this paper we discuss the convergence of distances associated to converging structures of Lipschitz vector fields and continuously varying norms on a smooth manifold. We prove that, under a mild controllability assumption on the limit vector-fields structure, the distances associated to equi-Lipschitz vector-fields structures that converge uniformly on compact subsets, and to norms that converge uniformly on compact subsets, converge locally uniformly to the limit Carnot-Carathéodory distance. In the case in which the limit distance is boundedly compact, we show that the convergence of the distances is uniform on compact sets. We show an example in which the limit distance is not…

Metric Lie groups admitting dilations

We consider left-invariant distances $d$ on a Lie group $G$ with the property that there exists a multiplicative one-parameter group of Lie automorphisms $(0, \infty)\rightarrow\mathtt{Aut}(G)$, $\lambda\mapsto\delta_\lambda$, so that $ d(\delta_\lambda x,\delta_\lambda y) = \lambda d(x,y)$, for all $x,y\in G$ and all $\lambda>0$. First, we show that all such distances are admissible, that is, they induce the manifold topology. Second, we characterize multiplicative one-parameter groups of Lie automorphisms that are dilations for some left-invariant distance in terms of algebraic properties of their infinitesimal generator. Third, we show that an admissible left-invariant distance on a Lie …

Nilpotent Groups and Bi-Lipschitz Embeddings Into L1

We prove that if a simply connected nilpotent Lie group quasi-isometrically embeds into an L1 space, then it is abelian. We reach this conclusion by proving that every Carnot group that bi-Lipschitz embeds into L1 is abelian. Our proof follows the work of Cheeger and Kleiner, by considering the pull-back distance of a Lipschitz map into L1 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-Lipsch…

Area of intrinsic graphs and coarea formula in Carnot Groups

AbstractWe consider submanifolds of sub-Riemannian Carnot groups with intrinsic $$C^1$$ C 1 regularity ($$C^1_H$$ C H 1 ). Our first main result is an area formula for $$C^1_H$$ C H 1 intrinsic graphs; as an application, we deduce density properties for Hausdorff measures on rectifiable sets. Our second main result is a coarea formula for slicing $$C^1_H$$ C H 1 submanifolds into level sets of a $$C^1_H$$ C H 1 function.

Topics in the geometry of non-Riemannian lie groups

A historical account on characterizations ofC1-manifolds in Euclidean spaces by tangent cones

Abstract A historical account on characterizations of C 1 -manifolds in Euclidean spaces by tangent cones is provided. Old characterizations of smooth manifold (by tangent cones), due to Valiron (1926, 1927) and Severi (1929, 1934) are recovered; modern characterizations, due to Gluck (1966, 1968) and Tierno (1997) are restated. All these results are consequences of the Four-cones coincidence theorem due to [1] .