Modulus of families of sets of finte perimeter and quasiconformal maps between metric spaces of globally Q-bounded geometry

2020 ◽  
Vol 69 (1) ◽  
pp. 265-294
Author(s):  
Rebekah Jones ◽  
Panu Lahti ◽  
Nageswari Shanmugalingam
Keyword(s):  
Metric Spaces ◽  
Quasiconformal Maps ◽  
Bounded Geometry

scholarly journals Quasiconformal maps in metric spaces with controlled geometry

1998 ◽  
Vol 181 (1) ◽  
pp. 1-61 ◽  
Author(s):  
Juha Heinonen ◽  
Pekka Koskela
Keyword(s):  
Metric Spaces ◽  
Quasiconformal Maps

scholarly journals Coarse infinite-dimensionality of hyperspaces of finite subsets

2021 ◽  
Author(s):  
Thomas Weighill ◽  
Takamitsu Yamauchi ◽  
Nicolò Zava
Keyword(s):  
Banach Space ◽  
Metric Space ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Dimensional Properties ◽  
Bounded Geometry ◽  
Infinite Dimensional

AbstractWe consider infinite-dimensional properties in coarse geometry for hyperspaces consisting of finite subsets of metric spaces with the Hausdorff metric. We see that several infinite-dimensional properties are preserved by taking the hyperspace of subsets with at most n points. On the other hand, we prove that, if a metric space contains a sequence of long intervals coarsely, then its hyperspace of finite subsets is not coarsely embeddable into any uniformly convex Banach space. As a corollary, the hyperspace of finite subsets of the real line is not coarsely embeddable into any uniformly convex Banach space. It is also shown that every (not necessarily bounded geometry) metric space with straight finite decomposition complexity has metric sparsification property.

scholarly journals Uniformization with Infinitesimally Metric Measures

2021 ◽  
Author(s):  
Kai Rajala ◽  
Martti Rasimus ◽  
Matthew Romney
Keyword(s):  
Metric Spaces ◽  
Doubling Measure ◽  
Quasiconformal Maps ◽  
Uniformization Theorem ◽  
Quasiconformal Map

AbstractWe consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces X homeomorphic to $${{\mathbb {R}}}^2$$ R 2 . Given a measure $$\mu $$ μ on such a space, we introduce $$\mu $$ μ -quasiconformal maps$$f:X \rightarrow {{\mathbb {R}}}^2$$ f : X → R 2 , whose definition involves deforming lengths of curves by $$\mu $$ μ . We show that if $$\mu $$ μ is an infinitesimally metric measure, i.e., it satisfies an infinitesimal version of the metric doubling measure condition of David and Semmes, then such a $$\mu $$ μ -quasiconformal map exists. We apply this result to give a characterization of the metric spaces admitting an infinitesimally quasisymmetric parametrization.

scholarly journals Regularity of solutions to the fractional Cheeger-Laplacian on domains in metric spaces of bounded geometry

2022 ◽  
Vol 306 ◽  
pp. 590-632
Author(s):  
Sylvester Eriksson-Bique ◽  
Gianmarco Giovannardi ◽  
Riikka Korte ◽  
Nageswari Shanmugalingam ◽  
Gareth Speight
Keyword(s):  
Metric Spaces ◽  
Regularity Of Solutions ◽  
Bounded Geometry

Crystallography in two-dimensional metric spaces*

1969 ◽  
Vol 130 (1-6) ◽  
pp. 277-303 ◽  
Author(s):  
Aloysio Janner ◽  
Edgar Ascher
Keyword(s):  
Metric Spaces ◽  
Two Dimensional

Solution of Volterra integral inclusion in b-metric spaces via new fixed point theorem

2016 ◽  
Vol 2017 (1) ◽  
pp. 17-30 ◽  
Author(s):  
Muhammad Usman Ali ◽  
◽  
Tayyab Kamran ◽  
Mihai Postolache ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Metric Spaces ◽  
Integral Inclusion

ON THE QUANTIFICATION OF UNIFORM PROPERTIES.PART 2

2001 ◽  
Vol 37 (1-2) ◽  
pp. 169-184
Author(s):  
B. Windels
Keyword(s):  
Metric Spaces ◽  
Uniform Spaces ◽  
Complete Metric Spaces ◽  
The Subject

In 1930 Kuratowski introduced the measure of non-compactness for complete metric spaces in order to measure the discrepancy a set may have from being compact.Since then several variants and generalizations concerning quanti .cation of topological and uniform properties have been studied.The introduction of approach uniform spaces,establishes a unifying setting which allows for a canonical quanti .cation of uniform concepts,such as completeness,which is the subject of this article.

Teichmuller Geometry

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Uniqueness Theorem ◽  
Existence Theorems ◽  
Quadratic Differentials ◽  
Complex Structures ◽  
Metric Geometry ◽  
Quasiconformal Maps ◽  
Hyperbolic Structures ◽  
Teichmüller Metric ◽  
Uniqueness And Existence ◽  
Teichmüller Mapping

This chapter focuses on the metric geometry of Teichmüller space. It first explains how one can think of Teich(Sɡ) as the space of complex structures on Sɡ. To this end, the chapter defines quasiconformal maps between surfaces and presents a solution to the resulting Teichmüller's extremal problem. It also considers the correspondence between complex structures and hyperbolic structures, along with the Teichmüller mapping, Teichmüller metric, and the proof of Teichmüller's uniqueness and existence theorems. The fundamental connection between Teichmüller's theorems, holomorphic quadratic differentials, and measured foliations is discussed as well. Finally, the chapter describes the Grötzsch's problem, whose solution is tied to the proof of Teichmüller's uniqueness theorem.

Sign in / Sign up

Export Citation Format

Share Document