scholarly journals Bisectors determining unique pairs of points in the bidisk

2018 ◽  
Vol 29 (03) ◽  
pp. 1850018
Author(s):  
Virginie Charette ◽  
Todd A. Drumm ◽  
Youngju Kim
Keyword(s):  
Metric Space ◽  
Symmetric Spaces ◽  
Metric Geometry ◽  
Unique Pair ◽  
Hyperbolic Complex ◽  
Usual Formula

Bisectors are equidistant hypersurfaces between two points and are basic objects in a metric geometry. They play an important part in understanding the action of subgroups of isometries on a metric space. In many metric geometries (spherical, Euclidean, hyperbolic, complex hyperbolic, to name a few) bisectors do not uniquely determine a pair of points, in the following sense: completely different sets of points share a common bisector. The above examples of this non-uniqueness are all rank [Formula: see text] symmetric spaces. However, generically, bisectors in the usual [Formula: see text] metric are such for a unique pair of points in the rank [Formula: see text] geometry [Formula: see text]. This result indicates the striking assertion that non-uniqueness of bisectors holds for “most” geometries.

Quasi-metric geometry

2014 ◽  
Author(s):  
◽  
Dan Brigham
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Linear Structure ◽  
Abstract Objects ◽  
Metric Geometry ◽  
Natural Structure ◽  
University Of Missouri ◽  
The University

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] Every time one sees |x-y|, one is looking at a specific metric acting on x and y, whatever they may happen to be, usually numbers or vectors. The notion of the distance between two objects is one of the most fundamental and ubiquitous in many branches of mathematics. A quasi-metric is a generalization of the familiar notion of metric. This dissertation examines what happens in this new setting of quasi-metrics. In particular, in the first chapter we introduce quasi-metrics, provide examples of them, then, given an arbitrary quasi-metric, develop a procedure which allows us to construct a better quasi-metric. Then we look at topological matters, such as openness and continuity. After that, we look at functions on abstract objects called groupoids, which is yet another step toward generality, since the objects we consider here contain the class of quasi-metrics. Dealing with groupoids is useful because it provides a natural structure into which quasi-metrics and quasi-norms fit. After these preliminary chapters, we then introduce linear structure, meaning the quasi-metrics studied are defined on sets in which one can add two points together, and multiply points by numbers, as this is not possible in an abstract set. Next we quantify smoothness of quasi-metric spaces, and throw in measures. For the first six chapters, we worked within a given quasi-metric space, assigning to points the distance between them. The seventh and final chapter deals with the "distance'' between two distinct quasi-metrics.

scholarly journals The metric geometry of singularity types

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Tamás Darvas ◽  
Eleonora Di Nezza ◽  
Hoang-Chinh Lu
Keyword(s):  
Equivalence Relation ◽  
Metric Space ◽  
Local Context ◽  
Metric Geometry ◽  
Positive Mass ◽  
Singularity Type ◽  
Compact Kähler Manifold ◽  
Ideal Sheaves ◽  
Monge Ampere ◽  
Multiplier Ideal

AbstractLet X be a compact Kähler manifold. Given a big cohomology class {\{\theta\}}, there is a natural equivalence relation on the space of θ-psh functions giving rise to {\mathcal{S}(X,\theta)}, the space of singularity types of potentials. We introduce a natural pseudo-metric {d_{\mathcal{S}}} on {\mathcal{S}(X,\theta)} that is non-degenerate on the space of model singularity types and whose atoms are exactly the relative full mass classes. In the presence of positive mass we show that this metric space is complete. As applications, we show that solutions to a family of complex Monge–Ampère equations with varying singularity type converge as governed by the {d_{\mathcal{S}}}-topology, and we obtain a semicontinuity result for multiplier ideal sheaves associated to singularity types, extending the scope of previous results from the local context.

scholarly journals Cauchy Symmetric Spaces with Point-countable cs-Networks

2016 ◽  
Vol 57 (1) ◽  
pp. 163-170
Author(s):  
Luong Quoc Tuyen
Keyword(s):  
Symmetric Space ◽  
Metric Space ◽  
Symmetric Spaces

Abstract In this paper, we prove that a Cauchy symmetric space has a point-countable cs-network if and only if it is a 1-sequence-covering compact-covering quotient π, s-image of a metric space; if and only if it is a sequence-covering quotient π, s-image of a metric space.

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.

On Asymmetric Distances

2013 ◽  
Vol 1 ◽  
pp. 200-231 ◽  
Author(s):  
Andrea C.G. Mennucci
Keyword(s):  
Total Variation ◽  
Calculus Of Variations ◽  
Distance Function ◽  
Metric Space ◽  
Metric Spaces ◽  
Geodesic Segment ◽  
Intrinsic Metric ◽  
Typical Application ◽  
Variation Formula

Abstract In this paper we discuss asymmetric length structures and asymmetric metric spaces. A length structure induces a (semi)distance function; by using the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempotent. As a typical application, we consider the length of paths defined by a Finslerian functional in Calculus of Variations. In the second part we generalize the setting of General metric spaces of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a geodesic segment (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of intrinsic metric space.

Sign in / Sign up

Export Citation Format

Share Document