Related papers: The visual angle metric and quasiregular maps
We show that the definition of parabolic-like map can be slightly modified, by asking $\partial \Delta$ to be a quasiarc out of the parabolic fixed point, instead of the dividing arcs to be $C^1$ on $[-1,0]$ and $[0,1]$.
These notes provide an exposition on obtaining the well-known standard results of quasiregular maps on Riemannian manifolds, given the corresponding theory in the Euclidean setting. We recall several different approaches to first-order…
A quasisymmetric graph is a curve whose projection onto a line is a quasisymmetric map. We show that this class of curves is related to solutions of the reduced Beltrami equation and to a generalization of the Zygmund class $\Lambda_*$.…
When matching parts of a surface to its whole, a fundamental question arises: Which points should be included in the matching process? The issue is intensified when using isometry to measure similarity, as it requires the validation of…
A relative Schottky set in a planar domain D is a subset of D obtained by removing from D open geometric discs whose closures are in D and are pairwise disjoint. In this paper we study quasisymmetric and related maps between relative…
We construct sense-preserving univalent harmonic mappings which map the unit disk onto a domain which is convex in the horizontal direction, but with varying dilatation. Also, we obtain minimal surfaces associated with such harmonic…
We construct a Banach space satisfying that the nearest point map (also called proximity mapping or metric projection) onto any compact and convex subset is continuous but not uniformly continuous. The space we construct is locally…
We explore the combination theorem for a group G splitting as a graph of relatively hyperbolic groups. Using the fine graph approach to relative hyperbolicity, we find short proofs of the relative hyperbolicity of G under certain…
We discuss the geometry of some arithmetic orbifolds locally isometric to a product of real hyperbolic spaces of dimension two and three, and prove that certain sequences of non-uniform orbifolds are convergent to this space in a geometric…
We study piecewise quasiconformal covering maps of the unit circle. We provide sufficient conditions so that a conjugacy between two such dynamical systems has a quasiconformal or David extension to the unit disk. Our main result…
Many inequality relations between real vector quantities can be succinctly expressed as "weak (sub)majorization" relations. We explain these ideas and apply them in several areas: angles between subspaces, Ritz values, and graph Laplacian…
Recently we have shown that the equivalence classes of metrics on the double of a metric space $X$ form an inverse semigroup. Here we define an inverse subsemigroup related to a family of isometric subspaces of $X$, which is more…
Suppose that $X$ and $Y$ are quasiconvex and complete metric spaces, that $G\subset X$ and $G'\subset Y$ are domains, and that $f: G\to G'$ is a homeomorphism. In this paper, we first give some basic properties of short arcs, and then we…
We give simple proofs of various versions of the Schwarz lemma for real valued harmonic functions and for holomorphic (more generally harmonic quasi\-re\-gu\-lar, shortly HQR) mappings with the strip codomain. Along the way using the…
We provide sharp estimates for the intrinsic distances of Finsler metrics with precise boundary estimates. These metrics include the Kobayashi-Hilbert metric near strongly convex points, the minimal metric near convex and strongly minimally…
Two definitions for the rectfiability of hypersurfaces in Heisenberg groups $\mathbb{H}^n$ have been proposed: one based on $\mathbb{H}$-regular surfaces, and the other on Lipschitz images of subsets of codimension-$1$ vertical subgroups.…
The exterior and interior Schwarzschild solutions are rewritten replacing the usual radial variable with an angular one. This allows to obtain some results otherwise less apparent or even hidden in other coordinate systems.
The inverse problem of diffraction theory in essence amounts to the reconstruction of the atomic positions of a solid from its diffraction image. From a mathematical perspective, this is a notoriously difficult problem, even in the…
The classical uniformization theorem states that any simply connected Riemann surface is conformally equivalent to the disk, the plane, or the sphere, each equipped with a standard conformal structure. We give a similar uniformization for…
Using an optimal containment approach, we quantify the asymmetry of convex bodies in $\mathbb{R}^n$ with respect to reflections across affine subspaces of a given dimension. We prove general inequalities relating these ''Minkowski…