25 research outputs found
Spaces of small metric cotype
Naor and Mendel's metric cotype extends the notion of the Rademacher cotype
of a Banach space to all metric spaces. Every Banach space has metric cotype at
least 2. We show that any metric space that is bi-Lipschitz equivalent to an
ultrametric space has infinimal metric cotype 1. We discuss the invariance of
metric cotype inequalities under snowflaking mappings and Gromov-Hausdorff
limits, and use these facts to establish a partial converse of the main result.Comment: 14 pages, the needed isometric inequality now derived from the
literature, rather than proved by hand; other minor typos and errors fixe
Bochner Partial Derivatives, Cheeger-Kleiner Differentiability, and Non-Embedding
Among all Poincar\'e inequality spaces, we define the class of Cheeger
fractals, which includes the sub-Riemannian Heisenberg group. We show that
there is no bi-Lipschitz embedding of any Cheeger fractal into any
Banach space with the following property: there exists a bounded Euclidean
domain such that for any Lipschitz mapping ,
the Bochner partial derivatives of exist and are integrable.
This extends and provides context for an important related result of Creutz and
Evseev
Weak contact equations for mappings into Heisenberg groups
Let k>n be positive integers. We consider mappings from a subset of
k-dimensional Euclidean space R^k to the Heisenberg group H^n with a variety of
metric properties, each of which imply that the mapping in question satisfies
some weak form of the contact equation arising from the sub-Riemannian
structure of the Heisenberg group. We illustrate a new geometric technique that
shows directly how the weak contact equation greatly restricts the behavior of
the mappings. In particular, we provide a new and elementary proof of the fact
that the Heisenberg group H^n is purely k-unrectifiable. We also prove that for
an open set U in R^k, the rank of the weak derivative of a weakly contact
mapping in the Sobolev space W^{1,1}_{loc}(U;R^{2n+1}) is bounded by almost
everywhere, answering a question of Magnani. Finally we prove that if a mapping
from U to H^n is s-H\"older continuous, s>1/2, and locally Lipschitz when
considered as a mapping into R^{2n+1}, then the mapping cannot be injective.
This result is related to a conjecture of Gromov.Comment: 28 page
Sharp differentiability results for the lower local Lipschitz constant and applications to non-embedding
We give a sharp condition on the lower local Lipschitz constant of a mapping from a metric space supporting a Poincaré inequality to a Banach space with the Radon–Nikodym property that guarantees differentiability at almost every point. We apply these results to obtain a non-embedding theorem for a corresponding class of mappings
Quasiconformal mappings that highly distort dimensions of many parallel lines
We construct a quasiconformal mapping of -dimensional Euclidean space, , that simultaneously distorts the Hausdorff dimension of a nearly
maximal collection of parallel lines by a given amount. This answers a question
of Balogh, Monti, and Tyson.Comment: 12 page
Steiner's formula in the Heisenberg group
Steiner's tube formula states that the volume of an ∈-neighborhood of a smooth regular domain in ℝn is a polynomial of degree n in the variable ∈ whose coefficients are curvature integrals (also called quermassintegrals). We prove a similar result in the sub-Riemannian setting of the first Heisenberg group. In contrast to the Euclidean setting, we find that the volume of an ∈-neighborhood with respect to the Heisenberg metric is an analytic function of ∈ that is generally not a polynomial. The coefficients of the series expansion can be explicitly written in terms of integrals of iteratively defined canonical polynomials of just five curvature terms