Introduces a new metric to measure deviation from hyperbolicity.
problem Measuring how much a metric space deviates from being hyperbolic.
method Defining the quasi-hyperbolicity constant and analyzing its properties.
result The quasi-hyperbolicity constant provides a measure of deviation from hyperbolicity.
Study compares hyperbolic and quasihyperbolic metrics in plane domains.
problem Comparing hyperbolic and quasihyperbolic metrics in plane domains.
method Analyzes metric spaces and boundaries of hyperbolic domains, proving equivalence and constructing counterexamples.
result Hyperbolic and quasihyperbolic metric spaces are quasiisometrically equivalent for finitely connected hyperbolic domains, but not in general.
Sharp bounds on hyperbolic metrics in Ptolemaic spaces are derived.
problem Finding sharp bounds on hyperbolic metrics in Ptolemaic spaces.
method Construction of metrics on open subsets of Ptolemaic spaces.
result Sharp parameter bounds for hyperbolic and strongly hyperbolic metrics are derived.
Study Gromov hyperbolic domains in Minkowski space, proving equivalence to boundary properties.
problem Investigate Gromov hyperbolic domains in Minkowski space.
method Explicit comparisons between metrics, dynamical arguments, and quasi-hyperbolic metric.
result Gromov hyperbolicity of convex, future complete domains is equivalent to stable acausality of the boundary.
Characterizes rigid and flexible hyperbolic cone metrics and billiards.
problem Understanding the rigidity and flexibility of hyperbolic cone metrics and their billiard dynamics.
method Characterization through Liouville currents and deformation spaces.
result Generic rigidity and parameterization of deformation spaces for flexible metrics.
A real valued function φ of one variable is called a metric transform if for every metric space (X,d) the composition dφ=φ∘d is also a metric on X. We give a complete characterization of the class of approximately nondecreasing, unbounded metric transforms φ such that the trans…
Explains visual metrics on hyperbolic space boundaries.
problem Understanding the geometry of hyperbolic spaces.
method Construction of visual metrics, quasisymmetries, and invariants.
result Detailed examples and applications of Gromov's round trees.
Constructs an asymptotic metric for moduli space of centred hyperbolic monopoles.
problem Analyzing the moduli space of centred hyperbolic monopoles.
method Point particle approximation and geodesic motion analysis.
result Obtains a hyperbolic analogue of negative mass Taub-NUT metric.
We introduce a new quasi-isometry invariant of metric spaces called the hyperbolic dimension, hypdim, which is a version of the Gromov's asymptotic dimension, asdim. The hyperbolic dimension is at most the asymptotic dimension, however, unlike the asymptotic dimension, the hyperbolic dimension of any Euclidean space R^…
We show the equivalence of several characterizations of relative hyperbolicity for metric spaces, and obtain extra information about geodesics in a relatively hyperbolic space. We apply this to characterize hyperbolically embedded subgroups in terms of nice actions on (relatively) hyperbolic spaces. We also study the d…
We develop a natural and geometric way to realize the hyperbolic plane as the moduli space of marked genus 1 Riemann surfaces. To do so, a metric is defined on the Teichmüller space of the torus, inspired by Thurston's Lipschitz metric for the case of hyperbolic surfaces. Based on extremal Lipschitz maps, the Teichmüll…
Cohomology defines hyperbolic spaces and their subgraphs.
problem Characterizing hyperbolic spaces and their subgraphs.
method Complete cohomological characterization using ℓ∞-cohomology. result Cohomology vanishing characterizes hyperbolicity and acylindrical hyperbolicity.
We study the geometry of nonrelatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with …
Statistical hyperbolicity proven for Teichmüller space.
problem Harmonic measures from random walks on mapping class groups.
method Proving statistical hyperbolicity using Teichmüller metric.
result Teichmüller space is statistically hyperbolic for certain harmonic measures.
Complete list of solvable BS groups' actions on hyperbolic spaces.
problem Characterizing actions of solvable Baumslag-Solitar groups on hyperbolic metric spaces.
method Complete enumeration and classification of actions up to equivalence.
result Finitely many equivalence classes of actions, each containing a point, tree, or hyperbolic plane action.
We study the moduli space of negatively curved metrics of a hyperbolic manifold.
New interpretation of discrete conformality using polyhedral convex hulls.
problem Understanding discrete conformality in 3D.
method Epstein-Penner convex hull construction and induced metrics.
result New bijections and interpretations of discrete conformality.
New gauge condition fixes L2 metric divergence in hyperbolic monopole spaces.
problem Divergence of L2 metric on hyperbolic monopole moduli spaces. method Alternative gauge-fixing condition inspired by supersymmetry.
result Resulting geometry is hyperbolic hyperkähler, analogous to Euclidean monopole spaces.
New metric on geodesic currents connects different surface genera.
problem Understanding geodesic currents on surfaces of varying genera.
method Introducing a new asymmetric metric on the space of projective filling geodesic currents.
result Metric spaces of projective filling geodesic currents for surfaces of different genera are not isometric.
This paper improves probabilistic latent models on hyperbolic spaces.
problem Uncertainty in predictions due to geodesics crossing low-data regions.
method Augmenting hyperbolic manifold with a pullback metric for probabilistic pullback metrics.
result Geodesics on pullback metric respect both geometry and data distribution, reducing uncertainty.
Asymptotic subcone of an unbounded metric space is another metric space, capturing the structure of the original space at infinity. In this paper we define a functional metric space S which is an asymptotic subcone of the hyperbolic plane. This space is a real tree branching at every its point. Moreover, it is a homoge…
Defines mass for non-smooth hyperbolic spaces using a modified flow.
problem Defining mass for non-smooth, asymptotically hyperbolic spaces.
method Normalized Ricci-DeTurck flow with scalar curvature lower bound.
result Mass function well-defined for continuous metrics.
Study on metrics on Teichmüller space of one-holed tori.
problem Examining metrics on Teichmüller space of one-holed tori.
method Constructing natural Lipschitz maps and proving metric coincidences.
result Lipschitz and curve metrics coincide on Teichmüller space of one-holed tori.
The Cayley hyperbolic space minimizes volume entropy among finite-volume metrics.
problem Volume entropy rigidity in Cayley hyperbolic spaces.
method Repairing a gap in the proof of volume entropy rigidity theorem.
result Cayley hyperbolic space minimizes volume entropy.
Abstract notes on hyperbolic surfaces and Teichmüller spaces.
problem Understanding the geometry of surfaces and Teichmüller spaces.
method Survey of results on stretch lines and Thurston's metric.
result Analogies between Thurston's metric and Teichmüller's metric.
Study projection in acylindrically hyperbolic groups, proving sublinear tracking and growth bounds.
problem Projection phenomena in acylindrically hyperbolic groups.
method Analyzing shortest projections in word metrics and hyperbolic spaces.
result Sublinear tracking of shortest projections and effective growth bounds.
This paper investigates the notion of learning user and item representations in non-Euclidean space. Specifically, we study the connection between metric learning in hyperbolic space and collaborative filtering by exploring Mobius gyrovector spaces where the formalism of the spaces could be utilized to generalize the m…
New Einstein metrics created by modifying hyperbolic infinity.
problem Creating new Einstein metrics.
method Perturbing conformal infinity of geometrically finite hyperbolic metrics and applying inverse function theorem.
result Construct new examples of Einstein metrics.
The paper constructs hyperbolic elements in multiple spaces.
problem Constructing hyperbolic elements in multiple Gromov-hyperbolic spaces.
method Explicit construction under minimal conditions.
result Set of simultaneously hyperbolic elements has strictly positive density.
Geodesic currents on hyperbolic surfaces have dual spaces that are metric trees.
problem Understanding the dual spaces of geodesic currents on hyperbolic surfaces.
method Analyzing the geometric properties of dual spaces, including their hyperbolicity and completeness.
result The dual spaces of geodesic currents are Gromov hyperbolic metric tree-graded spaces.
Central limit theorem for Green metrics on hyperbolic groups.
problem Proving a central limit theorem for Green metrics on hyperbolic groups.
method Proving a central limit theorem for Green metrics on hyperbolic groups using probability measures and ordering elements.
result Proved a central limit theorem for Green metrics on hyperbolic groups.
Teichmüller space rigidity proven for Thurston metric.
problem Understanding isometries in Teichmüller space with Thurston metric.
method Analyzing R-linear surjective isometries between cotangent spaces. result Every isometry between hyperbolic surfaces induces an isometry in Teichmüller space.
We investigate the rigidity of hyperbolic cone metrics on 3-manifolds which are isometric gluing of ideal and hyper-ideal tetrahedra in hyperbolic spaces. These metrics will be called ideal and hyper-ideal hyperbolic polyhedral metrics. It is shown that a hyper-ideal hyperbolic polyhedral metric is determined up to i…
The paper defines two types of hyperbolicity for complex manifolds and proves related results.
problem Defining and studying hyperbolicity for a broader class of complex manifolds.
method Introducing SKT hyperbolicity and Gauduchon hyperbolicity, proving results using SKT and Gauduchon metrics.
result Every SKT hyperbolic manifold is also Kobayashi/Brody hyperbolic and every Gauduchon hyperbolic manifold is divisorially hyperbolic.
Geodesics and boundaries found for metric structures on hyperbolic groups.
problem Understanding the space of metric structures on hyperbolic groups.
method Outer automorphism invariant geodesic bicombing and boundary construction.
result Boundary contains well-known pseudo metrics and rigidity results.
This paper classifies Ricci solitons in complex hyperbolic spaces.
problem Understanding Ricci solitons in complex hyperbolic spaces.
method Analyzing homogeneous expanding Ricci solitons as submanifolds of complex hyperbolic spaces.
result Classification and analysis of Lie subgroups with Ricci soliton induced metric in complex hyperbolic spaces.
Researchers create metrics on hyperbolic space's tangent bundle.
problem Constructing metrics on the unit tangent bundle of hyperbolic space.
method Using Hopf coordinates and Busemann functions, they constructed a flow-invariant metric.
result The unit tangent bundle of hyperbolic space is a homogeneous space under specific groups.
We study the geometry of hyperbolic cone surfaces, possibly with cusps or geodesic boundaries. We prove that any hyperbolic cone structure on a surface of non-exceptional type is determined up to isotopy by the geodesic lengths of a finite specific homotopy classes of non-peripheral simple closed curves. As an applicat…
Extended Otal-Peigné's Theorem to Gromov-hyperbolic spaces.
problem Prove critical exponent equals topological entropy for group actions.
method Extended Otal-Peigné's Theorem to proper, Gromov-hyperbolic spaces.
result Critical exponent equals topological entropy for line-convex spaces.
The main result is that every complete finite area hyperbolic metric on a sphere with punctures can be uniquely realized as the induced metric on the surface of a convex ideal polyhedron in hyperbolic 3-space. A number of other observations are included.
Study large deviations and speed of random walks in hyperbolic spaces.
problem Understanding the speed of random walks in hyperbolic spaces.
method Large deviations analysis for random walks with a non-elementary semi-group.
result Established large deviations results for random walk distances.
We investigate a generalization of the so-called metric splitting of globally hyperbolic space-times to non-smooth Lorentzian manifolds and show the existence of this metric splitting for a class of wave-type space-times. Our approach is based on smooth approximations of non-smooth space-times by families (or sequences…
Generalizes Rips' result on hyperbolic spaces to metric spaces, showing collapses for tree metrics.
problem Understanding the contractibility of Vietoris-Rips complexes in metric spaces.
method Extending Rips' result using geodesic defect and apparent pairs gradient.
result Vietoris-Rips complexes collapse to subforests for finite tree metrics.
Boundary rigidity defined for hyperbolic spaces, tied to geometric properties.
problem Understanding boundary rigidity in Gromov hyperbolic spaces.
method Analyzing properties of Gromov hyperbolic spaces and their boundaries.
result Boundary rigidity is equivalent to positive Cheeger isoperimetric constant and non-amenability.
Flat metrics on hyperbolic surfaces embed as polyhedral surfaces in (2+1)-spacetimes.
problem Embedding flat metrics on hyperbolic surfaces into (2+1)-spacetimes.
method Using convex polyhedral Cauchy surfaces and Teichmüller space properties.
result Existence and uniqueness of flat metrics embedding in (2+1)-spacetimes.
New interpretation of complex hyperbolic form as Weil-Petersson form.
problem Understanding complex hyperbolic structures on moduli spaces.
method Interpreting complex hyperbolic form as a Weil-Petersson form for punctured spheres.
result Found a new equality between two symplectic forms.
The paper classifies all left invariant metrics on complex hyperbolic space.
problem Classifying left invariant Riemannian metrics on complex hyperbolic space.
method Analyzing the structure of the Lie group and using properties of constant curvature metrics.
result All metrics are of constant negative scalar curvature, with only one Einstein.
Formula derived for volume entropy of certain metrics on Euclidean space.
problem Volume entropy calculation for a family of metrics.
method Derived a formula for volume entropy of metrics in a family of generalized SOL and hyperbolic space metrics.
result Solved a conjecture related to a family of 3-manifolds.