Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe visual angle metric and quasiregular maps
232
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The distortion of distances between points under maps is studied. We first prove a Schwarz-type lemma for quasiregular maps of the unit disk involving the visual angle metric. Then we investigate conversely the quasiconformality of a bilipschitz map with respect to the visual angle metric on convex domains. For the unit ball or half space, we prove that a bilipschitz map with respect to the visual angle metric is also bilipschitz with respect to the hyperbolic metric. We also obtain various inequalities relating the visual angle metric to other metrics such as the distance ratio metric and the quasihyperbolic metric.
rate research
Read More
A new similarity invariant metric $v_G$ is introduced. The visual angle metric $v_G$ is defined on a domain $GsubsetneqRn$ whose boundary is not a proper subset of a line. We find sharp bounds for $v_G$ in terms of the hyperbolic metric in the particular case when the domain is either the unit ball $Bn$ or the upper half space $Hn$. We also obtain the sharp Lipschitz constant for a Mobius transformation $f: Grightarrow G$ between domains $G$ and $G$ in $Rn$ with respect to the metrics $v_G$ and $v_{G}$. For instance, in the case $G=G=Bn$ the result is sharp.
We prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincare inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure spaces, in terms of directional derivatives in the direction of tangent vectors to suitable rectifiable curves.
In this paper we examine the relationship between hyperconvex hulls and metric trees. After providing a linking construction for hyperconvex spaces, we show that the four-point property is inherited by the hyperconvex hull, which leads to the theorem that every complete metric tree is hyperconvex. We also consider some extension theorems for these spaces.
We give study the Lipschitz continuity of Mobius transformations of a punctured disk onto another punctured disk with respect to the distance ratio metric.
The triangular ratio metric is studied in a domain $Gsubsetneqmathbb{R}^n$, $ngeq2$. Several sharp bounds are proven for this metric, especially, in the case where the domain is the unit disk of the complex plane. The results are applied to study the Holder continuity of quasiconformal mappings.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
