arXiv subjects

Manzi Huang

Publications and source records attributed to Manzi Huang.

24 records · Page 2Linked to original sources

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↗

arXiv2013

On the subinvariance of uniform domains in Banach spaces

Suppose that $E$ and $E'$ denote real Banach spaces with dimension at least 2, that $D\subset E$ and $D'\subset E'$ are domains, and that $f: D\to D'$ is a homeomorphism. In this paper, we prove the following subinvariance property for the class of uniform domains: Suppose that $f$ is a freely quasiconformal mapping and that $D'$ is uniform. Then the image $f(D_1)$ of every uniform subdomain $D_1$ in $D$ under $f$ is still uniform. This result answers an open problem of Väisälä in the affirmative.

math.MG↗

arXiv2012

Mapping problems for quasiregular mappings

We study images of the unit ball under certain special classes of quasiregular mappings. For homeomorphic, i.e., quasiconformal mappings problems of this type have been studied extensively in the literature. In this paper we also consider non-homeomorphic quasiregular mappings. In particular, we study (topologically) closed quasiregular mappings originating from the work of J. Väisälä and M. Vuorinen in 1970's. Such mappings need not be one-to-one but they still share many properties of quasiconformal mappings. The global behavior of closed quasiregular mappings is similar to the local behavior of quasiregular mappings restricted to a so-called normal domain.

math.CV↗

arXiv2012

On bilipschitz extensions in real Banach spaces

Suppose that $E$ and $E'$ denote real Banach spaces with dimension at least 2, that $D\not=E$ and $D'\not=E'$ are bounded domains with connected boundaries, that $f: D\to D'$ is an $M$-QH homeomorphism, and that $D'$ is uniform. The main aim of this paper is to prove that $f$ extends to a homeomorphism $\bar \bar{D}\to \bar{D}'$ and $\bar{f}|\partial D$ is bilipschitz if and only if $f$ is bilipschitz in $\bar{D}$. The answer to some open problem of Väisälä is affirmative under an natural additional condition.

math.CV↗

arXiv2012

On quasimöbius maps in real Banach spaces

Suppose that $E$ and $E'$ denote real Banach spaces with dimension at least 2, that $D\varsubsetneq E$ and $D'\varsubsetneq E'$ are domains, that $f: D\to D'$ is an $(M,C)$-CQH homeomorphism, and that $D$ is uniform. The main aim of this paper is to prove that $D'$ is a uniform domain if and only if $f$ extends to a homeomorphism $\bar{f}: \bar{D}\to \bar{D}'$ and $\bar{f}$ is $η$-QM relative to $\partial D$. This result shows that the answer to one of the open problems raised by Väisälä from 1991 is affirmative.

math.CV↗

arXiv2011

Quasihyperbolic geodesics in John domains in R^n

In this paper, we prove that if $D\subset R^n$ is a John domain which is homeomorphic to a uniform domain via a quasiconformal mapping, then each quasihyperbolic geodesic in $D$ is a cone arc, which shows that the answer to one of open problems raised by Heinonen in \cite{H} is affirmative. This result also shows that the answer to the open problem raised by Gehring, Hag and Martio in \cite{Gm} is positive for John domains which are homeomorphic to uniform domains via uasiconformal mappings. As an application, we prove that if $D\subset R^n$ is a John domain which is homeomorphic to a uniform domain, then $D$ must be a quasihyperbolic $(b, λ)$-uniform domain.

math.FA↗