We show that a relatively hyperbolic graph with uniformly hyperbolic peripheral subgraphs is hyperbolic. As an application, we show that the disc graph and the electrified disc graph of a handlebody H of genus g>1 are hyperbolic, and we determine their Gromov boundaries.
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48 results for “hyperbolic disc”
On the interplay between discrete invariants of complex hyperbolic disc bundles over surfacesmath.GT
Study on invariants of complex hyperbolic disc bundles over surfaces, proving a conjecture.
problem Investigating relationships between three invariants of complex hyperbolic disc orbibundles.
method Analyzing Euler characteristic, Euler number, and Toledo invariant of disc orbibundles over 2-orbifolds.
result Proved that -3|τ| = 2e + 2χ holds for certain complex hyperbolic disc orbibundles.
The paper describes p-adic analogs of hyperbolic discs and their geometric properties.
problem Exploring p-adic analogs of hyperbolic discs and their geometric properties.
method Geometric construction in the projective p-adic plane, using p-adic convexity and transitive action of PGL.
result Found three (or seven if p=2) p-adic analogs of hyperbolic discs, unlike the real case where only one exists.
We calculate the asymptotic average rate at which a generic geodesic on a finite area hyperbolic 2-orbifold returns to an embedded disc on the surface, as well as the average amount of time it spends in the disc during each visit. This includes the case where the center of the disc is a cone point.
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.
New approach to extremal hyperbolic surfaces using NEC groups.
problem Structural description of extremal hyperbolic surfaces.
method Uniformization by NEC groups for surfaces with cusps and/or geodesic boundary.
result Full description of automorphism groups of extremal surfaces.
The paper proves a Penrose inequality for 4-discs with hyperbolic ends and conic singularities.
problem Proving a Penrose inequality for conformal metrics on a unit 4-disc with hyperbolic ends and conic singularities.
method Defining a mass term and proving a Penrose type inequality with curvature condition.
result Proves a Penrose inequality for conformal metrics on a unit 4-disc with hyperbolic ends and conic singularities.
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.
Study complex hyperbolic structures on a disc orbibundle with 5 cone points.
problem Constructing complex hyperbolic structures on a disc orbibundle with vanishing Euler number.
method Analyzing involutions in and using bending-connectedness.
result A 4-dimensional bending-connected family of complex hyperbolic structures on a disc orbibundle.
Develops orbibundle theory for complex hyperbolic geometry.
problem Calculating invariants of complex hyperbolic disc orbibundles.
method Using diffeology, calculates Euler number and Toledo invariant.
result New tools for invariants of complex hyperbolic orbibundles.
The paper finds the Finsler structure of Apollonian weak metric on unit disc.
problem Understanding the Finsler structure of Apollonian weak metric on the unit disc.
method Analyzing the deformation of hyperbolic Poincaré metric by a closed 1-form.
result The Apollonian weak-Finsler structure has bounded below -curvature and flag curvature satisfying .
Minimal perimeter polygons in punctured discs are found with inscribed horocycles.
problem Finding polygons with minimal perimeter in punctured discs.
method Proving minimal perimeter by inscribed horocycles, generalizing to cone points and geodesic boundaries.
result Minimum perimeter polygons found with inscribed horocycles.
New complex structure on hyperbolic disc within hyperkaehler space.
problem Complex structure on hyperbolic disc within hyperkaehler space.
method Mostow decomposition and complex adjoint orbit analysis.
result Complex structure on hyperbolic disc differs from natural embedding.
Investigates properties of a pseudometric on domains in Euclidean space, linking it to hyperbolic geometry.
problem Defines and analyzes a pseudometric on domains in to understand their hyperbolic properties.
method Introduces a pseudometric based on conformal harmonic discs and studies its properties and conditions for hyperbolicity.
result Characterizes domains as hyperbolic based on their geometric properties and provides sufficient conditions for hyperbolicity.
The study shows that certain complex geometries are hyperbolic and contractible but fail to be CAT(0).
problem The failure of certain complex geometries to be CAT(0) despite being hyperbolic and contractible.
method The study uses combinatorial methods to demonstrate the failure of these geometries to satisfy a combinatorial isoperimetric inequality.
result The study proves that these geometries, while hyperbolic and contractible, do not satisfy a combinatorial isoperimetric inequality.
We study complex hyperbolic disc bundles over closed orientable surfaces that arise from discrete and faithful representations H_n->PU(2,1), where H_n is the fundamental group of the orbifold S^2(2,...,2) and thus contains a surface group as a subgroup of index 2 or 4. The results obtained provide the first complex hyp…
Sharp estimate on harmonic maps at conformal points in balls.
problem Estimating harmonic maps at conformal points in balls.
method Sharp estimate on differential norm using Schwarz-Pick lemma.
result Generalizes classical Schwarz-Pick lemma and gives optimal for .
Groups satisfy linear surface isoperimetric functions.
problem Isoperimetric functions for surface diagrams in hyperbolic groups.
method Analyzing word-hyperbolic groups and their surface diagrams.
result Linear isoperimetric functions for all surface types in hyperbolic groups.
Minimal discs count as knot invariants in hyperbolic 4-space.
problem Counting minimal discs with a given boundary in hyperbolic 4-space.
method Using J-holomorphic curves in twistor space to count minimal discs, considering singularities at infinity.
result The number of minimal discs with a given boundary is a knot invariant.
We present and discuss several old and new methods for mapping a circular disc to a square. In particular, we present analytical expressions for mapping each point (u,v) inside the circular disc to a point (x,y) inside a square region. Ideally, we want the mapping to be smooth and invertible. In addition, we put emphas…
Plumbing of surfaces embeds in hyperbolic 4-manifolds.
problem Embedding specific geometric structures in hyperbolic manifolds.
method Using plumbings of disc bundles over surfaces.
result Plumbed structures can be embedded in closed hyperbolic 4-manifolds.
Applying the techniques developed in [AGG], we construct new real hyperbolic manifolds whose underlying topology is that of a disc bundle over a closed orientable surface. By the Gromov-Lawson-Thurston conjecture [GLT], such bundles should satisfy the inequality , where stands for the E…
Tight embedding proves Gromov norm for quaternionic Kähler class.
problem Calculating the Gromov norm for quaternionic Kähler class.
method Embedding of quaternionic hyperbolic disc into quaternionic hyperbolic n-space.
result Value of the Gromov norm of the quaternionic Kähler class determined.
New hyperbolic metrics with isolated singularities constructed.
problem Constructing hyperbolic metrics with specific singularities.
method Correspondence between metrics and projective functions.
result Explicit construction of new hyperbolic metrics.
In this paper, we show that the nonexistence of rotationally symmetric harmonic diffeomorphism between the unit disk without the origin and a punctured disc with hyperbolic metric on the target.
We prove that the supremum of principal curvatures of a minimal embedded disc in hyperbolic three-space spanning a quasicircle in the boundary at infinity is estimated in a sublinear way by the norm of the quasicircle in the sense of universal Teichmüller space, if the quasicircle is sufficiently close to being the bou…
This article is based on the methods developed in [AGG]. We construct a complex hyperbolic structure on a trivial disc bundle over a closed orientable surface (of genus 2) thus solving a long standing problem in complex hyperbolic geometry (see [Gol1, p. 583] and [Sch, p. 14]). This example answers also [Eli, Open …
We construct a weakly complete flat surface in hyperbolic 3-space having a pair of hyperbolic Gauss maps both of whose images are contained in an arbitrarily given open disc in the ideal boundary of H^3. This construction is accomplished as an application of the minimal surface theory. This looks an interesting phenome…
Exploiting the geometry of heterogeneous networks: A case study of the Indian stock marketphysics.soc-ph
Study models Indian stock market using hyperbolic geometry for market stability and volatility analysis.
problem Identifying market stability and volatility in the Indian stock market.
method Modelled as a heterogeneous scale-free network, embedded in a 2D hyperbolic space, applied coalescent embedding, hyperbolic kmeans, and Bollinger Band analysis.
result Clusters in the embedded network better represent market communities than Euclidean clusters, allowing for early detection of market changes.
Study knot groups from disc patterns in 3D space.
problem Understanding groups generated by reflections in planar patterns.
method Construct and study representation spaces of holonomy groups of hyperbolic 3-manifolds.
result Visualizations suggest conjectures and support algebraic calculations.
The edge of torn elastic sheets and growing leaves often form a hierarchical buckling pattern. Within non-Euclidean plate theory this complex morphology can be understood as low bending energy isometric immersions of hyperbolic Riemannian metrics. With this motivation we study the isometric immersion problem in strip a…
Hyperbolic Flowersmath.HO
Crochet models show hyperbolic geometry properties.
problem Visualizing hyperbolic geometry concepts.
method Triangulated hyperbolic plane models.
result Demonstrates various hyperbolic geometry properties.
Computational method finds examples of extremal hyperbolic surfaces.
problem Constructing explicit examples of extremal hyperbolic surfaces is complicated.
method Brute force computational procedure.
result Examples of extremal hyperbolic surfaces constructed in all cases.
We study subgroups of generated by two non-commuting unipotent maps and whose product is also unipotent. We call the set of conjugacy classes of such groups. We provide a set of coordinates on that make it homeomorphic to . By considering the actio…
Geometry over algebrasmath.DG
Geometric structures over algebras describe geodesics and spaces.
problem Understanding geodesics in hyperbolic and Euclidean spaces over non-standard algebras.
method Study geometric structures from Hermitian forms on real algebras (dual numbers, split-complex, split-quaternions).
result Presented a projective model for the hyperbolic bidisc.
Given a closed hyperbolic 3-manifold M with a quasigeodesic flow we construct a π_1-equivariant sphere-filling curve in the boundary of hyperbolic space. Specifically, we show that any complete transversal P to the lifted flow on H^3 has a natural compactification as a closed disc that inherits a π_1 action. The embedd…
A geometric interpretation is given for certain elliptic-hyperbolic systems in the plane. Among several examples, one which reduces in the elliptic region to the equations for harmonic 1-forms on the projective disc is studied in detail. A boundary-value problem for this example is formulated and is shown to possess we…
If M is a hyperbolic 3-manifold with a quasigeodesic flow then we show that π_1(M) acts in a natural way on a closed disc by homeomorphisms. Consequently, such a flow either has a closed orbit or the action on the boundary circle is Möbius-like but not conjugate into PSL(2, R). We conjecture that the latter possibility…
We show some generic (robust) properties of smooth surfaces immersed in the real 3-space (Euclidean, affine or projective), in the neighbourhood of a {\em godron} (term due to R.Thom): an isolated parabolic point at which the (unique) asymptotic direction is tangent to the parabolic curve. With the help of these proper…
Linear progress in fibresmath.GT
Linear progress observed in fibered 3-manifold embeddings.
problem Understanding the geometric distortion of fibered embeddings.
method Verification of linear progress along typical rays using geometry and ergodic theory.
result A typical ray in the fiber makes linear progress in the ambient metric.
The paper studies the automorphism groups of specific 3-manifolds and finds upper bounds for their sizes.
problem Determining the size of automorphism groups of certain 3-manifolds.
method Analyzing the structure of MDC-Schottky extension groups for specific types of 3-manifolds.
result Upper bounds for the sizes of automorphism groups of specific 3-manifolds are derived and proven.
Smoothly isotopic 3-discs in 4-sphere become identical after 5-dimensional push.
problem Smooth isotopy of 3-discs in 4-sphere.
method Pushing 3-discs into 5-dimensional space.
result Isotopic 3-discs in 4-sphere become identical after 5-dimensional push.
The paper proves conditions for the existence of holomorphic discs in Kähler manifolds.
problem Existence of holomorphic discs for higher operations.
method Showing existence of minimal discs with specific properties implies existence of holomorphic discs.
result Minimal discs in Kähler manifolds with certain boundary conditions are holomorphic.
In this paper we develop a new theory of infinitesimal harmonic deformations for compact hyperbolic 3-manifolds with ``tubular boundary''. In particular, this applies to complements of tubes of radius at least $R_0 = \arctanh(1/\sqrt{3}) \approx 0.65848$ around the singular set of hyperbolic cone manifolds, removing th…
Hyperbolic geometry reveals financial network structure and systemic importance.
problem Understanding the structure and importance of financial networks.
method Data from European banking stress tests, hyperbolic geometry analysis.
result Latent dimensions of `popularity' and `similarity' are strongly associated with systemic importance.
Study on the topology of ordered disc configurations, revealing nontrivial homotopy classes.
problem Topology of ordered disc configurations and their homotopy types.
method Analysis of ordered configuration spaces of hard discs, focusing on homotopy types and nontrivial classes.
result Exhibit nontrivial classes in π_{n-3} for all n, and their persistence in deformed ambient discs.
Classifies homotopy ribbon discs for certain slice knots.
problem Characterizing homotopy ribbon discs for specific slice knots.
method Classifies -homotopy ribbon slice discs up to topological ambient isotopy.
result In the infinite cyclic case, there is a unique equivalence class of such slice discs. For the Baumslag-Solitar group, there are at most two equivalence classes of -homotopy ribbon discs.
New knots found with tough, unsliceable discs.
problem Finding tough knots that can't be sliced smoothly.
method Constructed infinitely many knots with non-approximable slice discs.
result Smoothly sliceable knots have non-approximable slice discs.