Related papers: Foliated corona decompositions

In the context of sub-Riemannian Heisenberg groups Hn, n \geq 1, we shall study Isoperimetric Profiles, which are closed compact hypersurfaces having constant horizontal mean curvature, very similar to ellipsoids. Our main goal is to study…

This paper studies bounds in a strong form of regularity for $3$-uniform hypergraphs which was developed by Frankl, Gowers, Kohayakawa, Nagle, R\"{o}dl, Skokan, and Schacht. Regular decompositions of this type involve two structural…

We construct and study the ideal Poisson--Voronoi tessellation of the product of two hyperbolic planes $\mathbb{H}_{2}\times \mathbb{H}_{2}$ endowed with the $L^{1}$ norm. We prove that its law is invariant under all isometries of this…

Let $\H$ denote the discrete Heisenberg group, equipped with a word metric $d_W$ associated to some finite symmetric generating set. We show that if $(X,\|\cdot\|)$ is a $p$-convex Banach space then for any Lipschitz function $f:\H\to X$…

We begin a coarse geometric study of Hilbert geometry. Actually we give a necessary and sufficient condition for the natural boundary of a Hilbert geometry to be a corona, which is a nice boundary in coarse geometry. In addition, we show…

In this paper, we consider homogeneous $\Delta_H$-harmonic polynomials on the first Heisenberg group $\mathbb H$ and their traces on the unit sphere $S_\rho$ associated with the Kor\'anyi--Folland homogeneous norm $\rho$. We prove that…

We present a strategy for proving an asymptotic upper bound on the number of defects (non-hexagonal Voronoi cells) in the $n$ generator optimal quantizer on a closed surface (i.e., compact 2-manifold without boundary). The program is based…

We prove that if $P,\mathcal{L}$ are finite sets of $\delta$-separated points and lines in $\mathbb{R}^{2}$, the number of $\delta$-incidences between $P$ and $\mathcal{L}$ is no larger than a constant times $$|P|^{2/3}|\mathcal{L}|^{2/3}…

The Heisenberg group $\mathbb{H}$ equipped with a sub-Riemannian metric is one of the most well known examples of a doubling metric space which does not admit a bi-Lipschitz embedding into any Euclidean space. In this paper we investigate…

We consider an asymmetric left-invariant norm $||\cdot ||_K$ in the first Heisenberg group $\mathbb{H}^1$ induced by a convex body $K\subset\mathbb{R}^2$ containing the origin in its interior. Associated to $\|\cdot\|_K$ there is a…

We provide a new and elementary proof for the structure of geodesics in the Heisenberg group $\mathbb{H}^n$. The proof is based on a new isoperimetric inequality for closed curves in $\mathbb{R}^{2n}$. We also prove that the…

Let $X$ be a Banach space or more generally a complete metric space admitting a conical geodesic bicombing. We prove that every closed $L$-Lipschitz curve $\gamma:S^1\rightarrow X$ may be extended to an $L$-Lipschitz map defined on the…

Kevin Hartshorn showed that if a three-dimensional manifold $M$ admits a Heegaard surface $\Sigma$ with Hempel distance $d$ then every incompressible surface in $M$ has genus at least $\frac{d}{2}$. Scharlemann-Tomova generalized this,…

We find sharp bounds for the norm inequality on a Pseudo-hermitian manifold, where the L^2 norm of all second derivatives of the function involving horizontal derivatives is controlled by the L^2 norm of the sub-Laplacian. Perturbation…

Let $M$ be a non-compact connected Riemann surface of finite type, and $R\subset\subset M$ be a relatively compact domain such that $H_{1}(M,\Z)=H_{1}(R,\Z)$. Let $\tilde R\longrightarrow R$ be a covering. We study the algebra…

Motivated by the Lawrence-Krammer-Bigelow representations of the classical braid groups, we study the homology of unordered configurations in an orientable genus-$g$ surface with one boundary component, over non-commutative local systems…

This paper is about the family of smooth quartic surfaces $X \subset \mathbb{P}^3$ that are invariant under the Heisenberg group $H_{2,2}$. For a very general such surface $X$, we show that the Picard number of $X$ is 16 and determine its…

We present a lower bound for a fragmentation norm and construct a bi-Lipschitz embedding $I\colon \mathbb{R}^n\to\mathrm{Ham}(M)$ with respect to the fragmentation norm on the group $\mathrm{Ham}(M)$ of Hamiltonian diffeomorphisms of a…

In this paper, we prove a generalization of a discreteness criteria for a large class of subgroups of PSL$_2(\mathbb{C})$. In particular, we show that for a given finitely generated, purely loxodromic, free Kleinian group…

A natural notion of higher order rectifiability is introduced for subsets of Heisenberg groups $\mathbb{H}^n$ in terms of covering a set almost everywhere by a countable union of $(\mathbf{C}_H^{1,\alpha},\mathbb{H})$-regular surfaces, for…