Introduces Möbius structures and hyperbolic ends for -surfaces in hyperbolic space.
problem Understanding -surfaces in hyperbolic space.
method Studies Möbius structures and hyperbolic ends.
result Applications to -surfaces in hyperbolic space.
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48 results for “hyperbolic structures”
By using Klein's model for hyperbolic geometry, hyperbolic structures on orbifolds or manifolds provide examples of real projective structures. By Andreev's theorem, many 3-dimensional reflection orbifolds admit a finite volume hyperbolic structure, and such a hyperbolic structure is unique. However, the induced real p…
Unique median structures found in hyperbolic spaces.
problem Uniqueness of median structures in hyperbolic spaces.
method Analyzing product of hyperbolic spaces and properties of relative hyperbolicity.
result Non-hyperbolic pants graphs can have unique median structures.
No spin structures found in a hyperbolic 4D space.
problem Finding hyperbolic 4-manifolds without spin structures.
method Constructed a non-compact, orientable, hyperbolic 4-manifold.
result Demonstrated existence of a hyperbolic 4D space without spin structures.
The study finds many tight contact structures on hyperbolic 3-spheres.
problem Finding tight contact structures on hyperbolic 3-spheres.
method Constructing hyperbolic homology 3-spheres and analyzing their tight contact structures.
result Produces hyperbolic homology 3-spheres with multiple distinct tight contact structures.
Compactifies CR structures for complex hyperbolic manifolds.
problem Building compact CR structures for complex hyperbolic manifolds.
method Constructs a compactification by a strictly pseudoconvex CR structure.
result Establishes a compact CR structure for asymptotically locally complex hyperbolic manifolds.
Infinite-genus surfaces have many isospectral hyperbolic structures.
problem Finding many isospectral hyperbolic structures on infinite-genus surfaces.
method Constructing families of isospectral hyperbolic structures on infinite-type surfaces without planar ends.
result Uncountable families of isospectral and quasiconformally distinct hyperbolic structures on infinite-genus surfaces with self-similar end spaces.
Characterizes boundaries of HHGs and their properties.
problem Understanding boundaries and properties of HHGs.
method Simplicial structure on hierarchically hyperbolic boundaries, modifications of HHG structures.
result Characterizes relative hyperbolicity and thickness of HHGs.
Classifies 3-manifold groups with equivariant hierarchically hyperbolic structures.
problem Classifying 3-manifold groups with equivariant hierarchically hyperbolic structures.
method Construction of suitable quasimorphisms on Seifert pieces to construct actions on quasi-lines.
result 3-manifold groups admit equivariant hierarchically hyperbolic structures.
Study on projective structures on a hyperbolic 3-orbifold using tetrahedra.
problem Analyzing projective structures on hyperbolic 3-orbifolds.
method Parameterization using classical invariants, traces, and geometric cross ratios.
result Computed and analyzed the moduli space of projective structures.
The paper applies combinatorial Ricci flows to prove hyperbolic structures on 3-manifolds.
problem Proving the existence of hyperbolic structures on 3-manifolds with cusps.
method Combinatorial Ricci curvature flow methods to study pseudo 3-manifolds and ideal triangulations.
result The extended Ricci flow converges to a decorated hyperbolic polyhedral metric if and only if there exists a zero Ricci curvature metric.
New examples of 5D manifolds without certain structures.
problem Finding manifolds without specific geometric structures.
method Relative strict hyperbolization.
result Construction of 5D manifolds without real projective or flat conformal structures.
Proves hyperbolic 3-manifolds have angle structures under certain conditions.
problem Proving angle structures for non-compact hyperbolic 3-manifolds.
method Subdividing ideal polyhedral decompositions and applying topological conditions.
result Proves existence of ideal triangulations with angle structures.
We consider a volume maximization program to construct hyperbolic structures on triangulated 3-manifolds, for which previous progress has lead to consider angle assignments which do not correspond to a hyperbolic metric on each simplex. We show that critical points of the generalized volume are associated to geometric …
We explain how to construct certain potential functions for the hyperbolic structures of a knot complement, which are closely related to the analytic functions on the deformation space of hyperbolic structures.
We classify hyperbolic monopoles with continuous symmetries and construct new examples.
problem Classifying and constructing hyperbolic monopoles with continuous symmetries.
method Developed a Structure Theorem and used representation theory to simplify the problem.
result Found constraints on structure groups and constructed novel spherically symmetric hyperbolic monopoles.
New combinatorial structure for hierarchically hyperbolic spaces.
problem Constructing new hierarchically hyperbolic spaces.
method Combinatorial hierarchical hyperbolicity criterion to construct and clarify HHS structures.
result HHSs admit a combinatorial structure, clarifying the application of the combinatorial HHS criterion.
We show the existence of tight contact structures on infinitely many hyperbolic three-manifolds obtained via Dehn surgeries along sections of hyperbolic surface bundles over circle.
New 2D complex hyperbolic structures found on sphere orbibundles.
problem Locally rigid complex hyperbolic structures on sphere orbibundles.
method Constructing families of complex hyperbolic structures on disc orbibundles.
result Examples of non-locally rigid complex hyperbolic structures.
Optimal discrete harmonic maps between hyperbolic surfaces are found via minimizing energy.
problem Finding optimal discrete harmonic maps between hyperbolic surfaces.
method Minimizing Dirichlet energy over all possible hyperbolic structures and realizations within a fixed homotopy class.
result At the optimal hyperbolic structure, the discrete harmonic map and edge weights are induced from a weighted Delaunay decomposition.
Researchers found only one hyperbolic structure for Borromean rings.
problem Characterizing hyperbolic structures in knot complements.
method Analyzing the fundamental group of the Borromean rings' complement and its representations in PSL(2,C).
result Borromean rings admit exactly one hyperbolic structure.
We consider the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic cone-manifold structure on a surface, with all interior cone angles being integer multiples of , determines a holonomy representation …
New discrete conformal structures on surfaces with boundary, proving global rigidity and constructing hyperbolic metrics.
problem Creating new discrete conformal structures on surfaces with boundary.
method Introducing new discrete conformal structures, proving global rigidity using variational principles, and introducing combinatorial curvature flows.
result Global rigidity of new discrete conformal structures and effective algorithms for constructing hyperbolic metrics.
Proposes HypCSE for enhanced hierarchical clustering.
problem Challenges in existing hierarchical clustering methods.
method Hyperbolic Continuous Structural Entropy (HypCSE) neural networks.
result Superior performance on seven datasets.
This notes explores angle structures on ideally triangulated compact -manifolds with high genus boundary. We show that the existence of angle structures implies the existence of a hyperbolic metric with totally geodesic boundary, and conversely each hyperbolic -manifold with totally geodesic boundary has an ideal…
First example of a hyperbolic 4-orbifold underlying .
problem Finding closed hyperbolic 4-orbifolds with symplectic underlying spaces.
method Realized as the underlying space of a closed hyperbolic 4-orbifold.
result First example of a closed hyperbolic 4-orbifold with symplectic underlying space.
Boundary properties of hyperbolic groups are invariant under a maximization procedure.
problem Proving boundary properties of hierarchically hyperbolic groups are invariant.
method Proving boundary invariance under a maximization procedure.
result Boundary properties of hierarchically hyperbolic groups are invariant under maximization.
Study of holonomy algebras and Kahler homogeneous structures in complex hyperbolic spaces.
problem Characterizing holonomy algebras and Kahler homogeneous structures in complex hyperbolic spaces.
method Detailed investigation of holonomy algebras and their actions, using canonical connections and Tricerri-Vanhecke's classification.
result Formula for all Kahler homogeneous structures on complex hyperbolic spaces and related holonomy properties.
The Exact Computation of A Real Hyperbolic Structure on the Complement of the Borromean Ringsmath.GT
Paper computes hyperbolic structure of Borromean rings complement.
problem Computing hyperbolic structures for link complements.
method Classical construction of Thurston's sense.
result Exact computation of hyperbolic structure for Borromean rings.
The Funk-Finsler structure is constructed in hyperbolic models, including the Klein unit disc.
problem Constructing Funk-Finsler structures in hyperbolic models.
method Using Finsler isometries and explicit computations, the Funk-Finsler structure is constructed in various hyperbolic models.
result The Funk-Finsler structure in the Klein unit disc is a Randers metric.
We consider the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic cone-manifold structure on a surface, with all interior cone angles being integer multiples of , determines a holonomy representation …
In his 1985 paper Sullivan sketched a proof of his structural stability theorem for differentiable group actions satisfying certain expansion-hyperbolicity axioms. In this paper we relax Sullivan's axioms and introduce a notion of "meandering hyperbolicity" for group actions on geodesic metric spaces. This generalizati…
In this paper we define a new invariant of the incomplete hyperbolic structures on a 1-cusped finite volume hyperbolic 3-manifold M, called the ortholength invariant. We show that away from a (possibly empty) subvariety of excluded values this invariant both locally parameterises equivalence classes of hyperbolic struc…
Paper computes G-index for specific hyperbolic manifolds.
problem Computing G-index for closed, spin, hyperbolic manifolds.
method Developed techniques to compute G-index for 2- or 4-manifolds.
result Computed G-index for Davis hyperbolic 4-manifold.
Paper presents a new method for learning hyperbolic representations using tree structures.
problem Learning faithful low-dimensional hyperbolic embeddings of data.
method Metric-first approach to learn tree structure, then embed into hyperbolic manifold.
result Novel fast algorithm TreeRep learns tree approximating original metric.
This chapter from the upcoming Handbook of Knot Theory (eds. Menasco and Thistlethwaite) shows how to construct hyperbolic structures on link complements and perform hyperbolic Dehn filling. Along with a new elementary exposition of the standard ideas from Thurston's work, the article includes never-before-published ex…
Classifies tight contact structures on surgeries of the Whitehead link.
problem Classifying tight contact structures on surgeries of the Whitehead link.
method Analyzes various surgeries on the Whitehead link to classify tight contact structures.
result Determines tight contact structures, Stein fillability, and virtually overtwisted properties.
This paper introduces an acceleration structure for hyperbolic embeddings.
problem Efficiently embedding and visualizing high-dimensional data in hyperbolic spaces.
method Building upon a polar quadtree, the paper introduces a new acceleration structure for hyperbolic embeddings.
result The new method computes embeddings in significantly less time compared to existing methods.
Euclidean volumes of hyperbolic knots are algebraic numbers.
problem Understanding the algebraic nature of Euclidean volumes in hyperbolic knots.
method Deforming hyperbolic structures into Euclidean structures and analyzing the normalised Euclidean volumes.
result Normalised Euclidean volumes of hyperbolic knots are always algebraic numbers.
We show that the hyperbolic structure on a closed, orientable, hyperbolic 3-manifold can be constructed from a solution to the hyperbolic gluing equations using any triangulation with essential edges. The key ingredients in the proof are Thurston's spinning construction and a volume rigidity result attributed by Dunfie…
For 3-dimensional hyperbolic cone structures with cone angles , local rigidity is known for , but global rigidity is known only for . The proof of the global rigidity by Kojima is based on the fact that hyperbolic cone structures with cone angles at most do not degenerate in defo…
We characterize the class of Gromov hyperbolic spaces, whose boundary at infinity allow canonical Möbius structures.
Study finds optimal loops in hyperbolic space with Finsler structure.
problem Optimal loops in Finsler hyperbolic plane.
method Left-invariant Finsler structure, convex trigonometry functions.
result Optimal isoperimetric loops found in terms of trigonometry functions.
Whether every hyperbolic 3-manifold admits a tight contact structure or not is an open question. Many hyperbolic 3-manifolds contain taut foliations and taut foliations can be perturbed to tight contact structures. The first examples of hyperbolic 3-manifolds without taut foliations were constructed by Roberts, Sharesh…
Geometrically describes hyperbolic structures on link complements using quantum groups.
problem Describing hyperbolic structures on link complements algebraically.
method Uses octahedral decomposition and Kashaev-Reshetikhin's braiding on quantum group .
result Shows how to interpret geometrically the algebraic gluing equations for hyperbolic structures.
We advocate the use of cluster algebras and their y-variables in the study of hyperbolic 3-manifolds. We study hyperbolic structures on the mapping tori of pseudo-Anosov mapping classes of punctured surfaces, and show that cluster y-variables naturally give the solutions of the edge-gluing conditions of ideal tetrahedr…
New hyperbolicity concepts expand manifold study.
problem Studying hyperbolicity on complex manifolds.
method Introducing sG-hyperbolicity, weakly p-Kähler hyperbolic structures, and pluriclosed star split hyperbolic metrics.
result Expands the class of divisorially hyperbolic manifolds.
New method for computing hyperbolic structures on 3-manifolds with torus boundaries.
problem Computing a complete hyperbolic structure on 3-manifolds with torus boundaries.
method Convex optimization and combinatorial modifications to find a triangulation that admits a solution to the gluing equations.
result Experimental results support the new method for modifying triangulations and updating their geometry.