arXiv subjects

Qingshan Zhou

Publications and source records attributed to Qingshan Zhou.

29 records · Page 2Linked to original sources

arXiv2020

Quasimöbius invariance of uniform domains

In this paper, we study quasimöbius invariance of uniform domains in Banach spaces. We first investigate implications of certain geometric properties of domains in Banach spaces, such as the (diameter) uniformity, the $δ$-uniformity and the min-max property. Then we show that all of these conditions are equivalent if the domain is $ψ$-natural. As applications, we answer partially to an open question proposed by Väisälä, and provide a new method to prove a recent result of M. Huang, Y. Li, M. Vuorinen, and X. Wang in [On quasimöbius maps in real Banach spaces, Israel J. Math. 198 (2013), 467-486], which also gives an answer to another question raised by Väisälä.

math.CV↗

arXiv2020

The Gehring-Hayman type theorems on complex domains

In this paper we establish Gehring-Hayman type theorems for some complex domains. Suppose that $Ω\subset \mathbb{C}^n$ is a bounded $m$-convex domain with Dini-smooth boundary, or a bounded strongly pseudoconvex domain with $C^2$-smooth boundary. Then we prove that the Euclidean length of Kobayashi geodesic $[x,y]$ in $Ω$ is less than $c_1|x-y|^{c_2}$. Furthermore, if $Ω$ endowed with the Kobayashi metric is Gromov hyperbolic, then we can generalize this result to quasi-geodesics with respect to Bergman metric, Carathéodory metric or Kähler-Einstein metric. As applications, we prove the bi-Hölder equivalence between the Euclidean boundary and the Gromov boundary. Moreover, by using this boundary correspondence, we can show some extension results for biholomorphisms, and more general rough quasi-isometries with respect to the Kobayashi metrics between the domains.

math.CV↗

arXiv2020

Sphericalization and flattening with their applications in quasimetric measure spaces

The main purpose of the note is to explore the invariant properties of sphericalization and flattening and their applications in quasi-metric spaces. We show that sphericalization and flattening procedures on a quasimetric spaces preserving properties such as Ahlfors regular and doubling property. By using these properties, we generalize a recent result in \cite{WZ}. We also show that the Loewner condition can be preserved under quasimöbius mapping between two $Q$-Ahlfors regular spaces. Finally, we prove that the $Q$-regularity of $Q$-dimensional Hausdorff measure of Bourdon metric are coincided with Hausdorff measure of Hamenstädt metric defined on the boundary at infinity of a Gromov hyperbolic space.

math.CV↗

arXiv2019

Gromov hyperbolicity, John spaces and quasihyperbolic geodesics

We show that every quasihyperbolic geodesic in a John space admitting a roughly starlike Gromov hyperbolic quasihyperbolization is a cone arc. This result provides a new approach to the elementary metric geometry question, formulated in \cite[Question 2]{Hei89}, which has been studied by Gehring, Hag, Martio and Heinonen. As an application, we obtain a simple geometric condition connecting uniformity of the space with the existence of Gromov hyperbolic quasihyperbolization.

math.CV↗

arXiv2019

Deformations on symbolic Cantor sets and ultrametric spaces

By introducing new deformations on symbolic Cantor sets and ultrametric spaces, we prove that doubling ultrametric spaces admit bilipschitz embedding into Cantor sets. If in addition the spaces are uniformly perfect, we show that they are quasisymmetrically equivalent to Cantor sets. As an application, we provide a new proof for a recent work of Heer (Anal. Geom. Metr. Spaces, 2017) regarding quasimöbius uniformization of Cantor set.

math.CV↗

arXiv2019

Uniform perfectness for quasi-metric spaces

The aim of this paper is to investigate the equivalence conditions for uniform perfectness of quasi-metric spaces. We also obtain the invariant property of uniform perfectness under quasimöbius maps in quasi-metric spaces. In the end, two applications are given.

math.CV↗

arXiv2018

Weakly quasisymmetric maps and uniform spaces

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 show that: if $f$ is a weakly quasisymmetric mapping and $G'$ is a quasiconvex domain, then the image $f(D)$ of every uniform subdomain $D$ in $G$ is uniform. As an application, we get that if $f$ is a weakly quasisymmetric mapping and $G'$ is an uniform domain, then the images of the short arcs in $G$ under $f$ are uniform arcs in the sense of diameter.

math.CV↗

arXiv2018

Characterizations of John spaces

The main purpose of this paper is to study the characterizations of John spaces. We obtain five equivalence characteristics for length John spaces. As an application, we establish a dimension-free quasisymmetric invariance of length John spaces.This result is new also in the case of the Euclidean space.

math.CV↗

arXiv2018

Apollonian metric, uniformity and Gromov hyperbolicity

The main purpose of this paper is to investigate the properties of a mapping which is required to be roughly bilipschitz with respect to the Apollonian metric (roughly Apollonian bilipschitz) of its domain. We prove that under these mappings the uniformity, $φ$-uniformity and $δ$-hyperbolicity (in the sense of Gromov with respect to quasihyperbolic metric) of proper domains of $\mathbb{R}^n$ are invariant. As applications, we give four equivalent conditions for a quasiconformal mapping which is defined on a uniform domain to be roughly Apollonian bilipschitz, and we conclude that $φ$-uniformity is invariant under quasimöbius mappings.

math.CV↗

arXiv2015

Semisolidity and locally weak quasisymmetry of homeomorphisms in metric spaces

In this paper, we investigate the relationship between semisolidity and locally weak quasisymmetry of homeomorphisms in quasiconvex and complete metric spaces. Our main objectives are to (1) generalize the main result in [X. Huang and J. Liu, Quasihyperbolic metric and quasisymmetric mappings in metric spaces, Trans. Amer. Math. Soc. 367 (2015), 6225-6246] together with other related results, and (2) give a complete answer to the open problem given in [X. Huang and J. Liu, Quasihyperbolic metric and quasisymmetric mappings in metric spaces, Trans. Amer. Math. Soc. 367 (2015), 6225-6246]. As an application, we prove that the composition of two locally weakly quasisymmetric mappings is a locally weakly quasisymmetric mapping and that it is quasiconformal.

math.CV↗

arXiv2015

On the subinvariance of uniform domains in metric spaces

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. Our main result is the following subinvariance property of the class of uniform domains: Suppose both $f$ and $f^{-1}$ are weakly quasisymmetric mappings and $G'$ is a quasiconvex domain. Then the image $f(D)$ of every uniform subdomain $D$ in $G$ under $f$ is uniform. The subinvariance of uniform domains with respect to freely quasiconformal mappings or quasihyperbolic mappings is also studied with the additional condition that both $G$ and $G'$ are locally John domains.

math.CV↗