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Jeremy T. Tyson

Publications and source records attributed to Jeremy T. Tyson.

29 records · Page 2Linked to original sources
arXiv2016

Sub-Riemannian curvature and a Gauss-Bonnet theorem in the Heisenberg group

We use a Riemannnian approximation scheme to define a notion of $\textit{sub-Riemannian Gaussian curvature}$ for a Euclidean $C^{2}$-smooth surface in the Heisenberg group $\mathbb{H}$ away from characteristic points, and a notion of $\textit{sub-Riemannian signed geodesic curvature}$ for Euclidean $C^{2}$-smooth curves on surfaces. These results are then used to prove a Heisenberg version of the Gauss-Bonnet theorem. An application to Steiner's formula for the Carnot-Carathéodory distance in $\mathbb{H}$ is provided.

math.DG↗
arXiv2014

Marstrand's density theorem in the Heisenberg group

We prove that if $μ$ is a Radon measure on the Heisenberg group $\mathbb{H}^n$ such that the density $Θ^s(μ,\cdot)$, computed with respect to the Korányi metric $d_H$, exists and is positive and finite on a set of positive $μ$ measure, then $s$ is an integer. The proof relies on an analysis of uniformly distributed measures on $(\mathbb{H}^n,d_H)$. We provide a number of examples of such measures, illustrating both the similarities and the striking differences of this sub-Riemannian setting from its Euclidean counterpart.

math.MG↗
arXiv2013

Homotopy groups of spheres and Lipschitz homotopy groups of Heisenberg groups

We provide a sufficient condition for the nontriviality of the Lipschitz homotopy group of the Heisenberg group, $π_m^{Lip}(H_n)$, in terms of properties of the classical homotopy group of the sphere, $π_m(S^n)$. As an application we provide a new simplified proof of the fact that $π_n^{Lip}(H_n)\neq 0$, $n=1,2,...$, and we prove a new result that $π_{4n-1}^{Lip}(H_{2n})\neq 0$ for $n=1,2,...$ The last result is based on a new generalization of the Hopf invariant. We also prove that Lipschitz mappings are not dense in the Sobolev space $W^{1,p}(M,H_{2n})$ when $dim M\geq 4n$ and $4n-1\leq p<4n$.

math.GT↗
arXiv2013

Removable sets for Lipschitz harmonic functions on Carnot groups

Let G be a Carnot group with homogeneous dimension Q larger than 2 and let L be a sub-Laplacian on G. We prove that the critical dimension for removable sets of Lipschitz L-harmonic functions is Q-1. Moreover we construct self-similar sets with positive and finite (Q-1) dimensional Hausdorff measure which are removable.

math.AP↗
arXiv2013

Dimension distortion by Sobolev mappings in foliated metric spaces

We quantify the extent to which a supercritical Sobolev mapping can increase the dimension of subsets of its domain, in the setting of metric measure spaces supporting a Poincaré inequality. For foliations of a metric space X defined by a David--Semmes regular mapping we quantitatively estimate, in terms of Hausdorff dimension, the size of the set of leaves of the foliation that are mapped onto sets of higher dimension. We discuss key examples of such foliations, including foliations of the Heisenberg group by left and right cosets of horizontal subgroups. Examples are given demonstrating the extent to which our results are sharp. In fact, we show that the set of mappings that distort the dimensions of sets by the maximum possible amount is a prevalent subset of the relevant function space; the latter result is new even in Euclidean space.

math.MG↗
arXiv2013

Removable sets for homogeneous linear PDE in Carnot groups

Let $\cL$ be a homogeneous left invariant differential operator on a Carnot group. Assume that both $\cL$ and $\cL^t$ are hypoelliptic. We study the removable sets for $\cL$-solutions. We give precise conditions in terms of the Carnot--Carathéodory Hausdorff dimension for the removability for $\cL$-solutions under several auxiliary integrability or regularity hypotheses. In some cases, our criteria are sharp on the level of the relevant Hausdorff measure. One of the main ingredients in our proof is the use of novel local self similar tilings in Carnot groups.

math.AP↗
arXiv2013

Frequency of Sobolev dimension distortion of horizontal subgroups of Heisenberg groups

We study the behavior of Sobolev mappings defined on the Heisenberg groups with respect to a foliation by left cosets of a horizontal homogeneous subgroup. We quantitatively estimate, in terms of Euclidean Hausdorff dimension, the size of the set of cosets that are mapped onto sets of high dimension. The proof of our main result combines ideas of Gehring and Mostow about the absolute continuity of quasiconformal mappings with Mattila's projection and slicing machinery.

math.MG↗
arXiv2013

Modulus and Poincaré inequalities on non-self-similar Sierpinski carpets

A carpet is a metric space homeomorphic to the Sierpinski carpet. We characterize, within a certain class of examples, non-self-similar carpets supporting curve families of nontrivial modulus and supporting Poincaré inequalities. Our results yield new examples of compact doubling metric measure spaces supporting Poincaré inequalities: these examples have no manifold points, yet embed isometrically as subsets of Euclidean space.

math.MG↗
arXiv2012

On transversal submanifolds and their measure

We study the class of transversal submanifolds. We characterize their blow-ups at transversal points and prove a negligibility theorem for their "generalized characteristic set", with respect to the Carnot-Carathéodory Hausdorff measure. This set is made by all points of non-maximal degree. Observing that C^1 submanifolds in Carnot groups are generically transversal, the previous results prove that the "intrinsic measure" of C^1 submanifolds is generically equivalent to their Carnot-Carathéodory Hausdorff measure. As a result, the restriction of this Hausdorff measure to the submanifold can be replaced by a more manageable integral formula, that should be seen as a "sub-Riemannian mass". Another consequence of these results is an explicit formula, only depending on the embedding of the submanifold, that computes the Carnot-Carathéodory Hausdorff dimension of C^1 transversal submanifolds.

math.MG↗
arXiv2008

Helical CR Structures and Sub-Riemannian Geodesics

A helical CR structure is a decomposition of a real Euclidean space into an even-dimensional horizontal subspace and its orthogonal vertical complement, together with an almost complex structure on the horizontal space and a marked vector in the vertical space. We prove an equivalence between such structures and step two Carnot groups equipped with a distinguished normal geodesic, and also between such structures and smooth real curves whose derivatives have constant Euclidean norm. As a consequence, we relate step two Carnot groups equipped with sub-Riemannian geodesics with this family of curves. The restriction to the unit circle of certain planar homogeneous polynomial mappings gives an instructive class of examples. We describe these examples in detail.

math.CV↗