arXiv research

Surveying 33 mapping questions posed by Heinonen and Semmes.

In this paper we shall show that the boundary $\partial I_{p,q}$ of the hyperbolic building $I_{p,q}$ considered in M. Bourdon, \emph{Immeubles hyperboliques, dimension conforme et rigidité de Mostow} (Geometric And Functional Analysis, Vol 7 (1997), p 245-268) admits Poincaré type inequalities. Then by using Heinonen-…

We study metric spaces homeomorphic to the 2-sphere, and find conditions under which they are quasisymmetrically homeomorphic to the standard 2-sphere. As an application of our main theorem we show that an Ahlfors 2-regular, linearly locally contractible metric 2-sphere is quasisymmetrically homeomorphic to the standar…

Defines signed quasiregular curves and proves growth theorem.

We show that a closed, connected and orientable Riemannian manifold of dimension $d$ that admits a quasiregular mapping from $\mathbb R^d$ must have bounded cohomological dimension independent of the distortion of the map. The dimension of the degree $l$ de Rham cohomology of $M$ is bounded above by $\binom{d}{l}$. Thi…

A fundamental theorem of Wolfe isometrically identifies the space of flat differential forms of dimension $m$ in $\mathbb{R}^n$ with the space of flat $m$-cochains, that is, the dual space of flat chains of dimension $m$ in $\mathbb{R}^n$. The main purpose of the present paper is to generalize Wolfe's theorem to the se…

We consider decomposition spaces $\R^3/G$ that are manifold factors and admit defining sequences consisting of cubes-with-handles. Metrics on $\R^3/G$ constructed via modular embeddings into Euclidean spaces promote the controlled topology to a controlled geometry. The quasisymmetric parametrizability of the metric spa…

The aim of the present paper is to define a notion of weakly differentiable cochain in the generality of metric measure spaces and to study basic properties of such cochains. Our cochains are (sub-)linear functionals on a subspace of chains, and a suitable notion of chains in metric spaces is given by Ambrosio-Kirchhei…

The study extends removability results for quasiregular curves in Euclidean spaces.

In this article we prove that, for an oriented PL $n$-manifold $M$ with $m$ boundary components and $d_0\in \mathbb N$, there exist mutually disjoint closed Euclidean balls and a $\mathsf K$-quasiregular mapping $M \to \mathbb S^n \setminus \mathrm{int}(B_1\cup \cdots \cup B_m)$ of degree at least $d_0$. The result is …