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.
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.
Introduces Möbius structures and hyperbolic ends for k-surfaces in hyperbolic space.
problem Understanding k-surfaces in hyperbolic space. method Studies Möbius structures and hyperbolic ends.
result Applications to k-surfaces in hyperbolic space. Introduces hyperbolic generalized framed surfaces and their properties.
problem None explicitly stated; focuses on introducing new geometric objects.
method Generalization of hyperbolic framed surfaces and curves.
result Established conditions for a surface to be a hyperbolic generalized framed base surface and explored their singularities.
Study on new hyperbolicity notions for non-Kähler manifolds and their deformations.
problem Analyzing new hyperbolicity notions for non-Kähler complex manifolds.
method Introducing and analyzing two new notions of hyperbolicity for compact complex non-Kähler manifolds, and studying their behavior under smooth modifications.
result Established openness results for p-HS hyperbolicity and p-Kähler hyperbolicity under holomorphic deformations. New formulas for hyperbolic mass using horospheres.
problem Mass calculation of asymptotically hyperbolic manifolds.
method Geometric formulas derived using coordinate horospheres.
result Improved rigidity results of hyperbolic space.
Abstract: Non-residually finite hyperbolic groups imply non-residually finite rigid hyperbolic groups.
problem Existence of non-residually finite hyperbolic groups
method Direct implication
result Existence of non-residually finite rigid hyperbolic groups
Study complex hyperbolic lattices and their relation to strict hyperbolization.
problem Understanding the relationship between complex hyperbolic lattices and strict hyperbolization.
method Analyzing the fundamental groups of complex hyperbolic manifolds and spaces arising from strict hyperbolization.
result Uniform lattices in PU(n,1) cannot be fundamental groups of Charney-Davis strict hyperbolizations when n ≥ 2.
Abstract reviews hyperbolic positive energy theorems.
problem Analyzing positive energy theorems for hyperbolic spaces.
method Review of existing literature on asymptotically hyperbolic manifolds.
result Summarizes positive energy theorems for hyperbolic spaces.
We study hyperbolic bongles and find their volumes.
problem Characterizing and quantifying hyperbolic bongles.
method Provided necessary and sufficient conditions for hyperbolicity, calculated volumes, and established upper bounds.
result All balanced hyperbolic n-bongles have the same volume and this volume is an upper bound for any hyperbolic n-bongle. Paper finds inequalities for convex domains in hyperbolic space.
problem Finding inequalities for convex domains in hyperbolic space.
method Introducing hyperbolic ellipsoids and using orthogonal projection to establish inequalities.
result Affine isoperimetric inequalities for static convex domains in hyperbolic space characterized by hyperbolic ellipsoids.
We study conservative partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds. We show that they are always accessible and deduce as a result that every conservative C1+ partially hyperbolic in a hyperbolic 3-manifold must be ergodic, giving an afirmative answer to a conjecture of Hertz-Hertz-Ures in the co…
Study deformations of compact Kähler hyperbolic manifolds.
problem Investigate the deformation openness of Kähler hyperbolicity.
method Propose modified versions of Kähler hyperbolicity as a tool.
result Provide a first step towards understanding deformations.
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.
Study shows hyperbolic subgroups can be free products of surface and free groups.
problem Characterizing hyperbolic subgroups within larger groups.
method Analyzing fiber bundles and using properties of hyperbolic groups.
result Non-elementary hyperbolic commensurated subgroups are virtually free products of surface and free groups.
New 5D hyperbolic shapes that wrap around a circle found.
problem Finding new 5D hyperbolic structures.
method Examined finite-volume cusped hyperbolic 5-manifolds.
result Discovered the smallest known hyperbolic 5-manifold.
Study properties of hyperbolic Yamabe solitons in submanifolds.
problem Characterize and understand hyperbolic Yamabe solitons in submanifolds.
method Define hyperbolic Yamabe flow, prove properties, and classify solitons.
result Prove that hyperbolic Yamabe solitons are pseudosymmetric or metallic shaped.
Survey solves curvature problems with hyperbolic spaces.
problem Singularities in hypersurface geometry.
method Hyperbolic unfolding correspondence linking hypersurfaces to Gromov hyperbolic spaces.
result Eliminates hypersurface singularities in scalar curvature geometry.
Drilling hyperbolic groups to simplify complex conjectures.
problem Proving the Cannon Conjecture for hyperbolic groups with 2-sphere boundary.
method Defining drilling of hyperbolic groups and proving it preserves relative hyperbolicity.
result Reduction of the Cannon Conjecture to a more tractable relative version.
HKConv learns hyperbolic features by aggregating kernel points.
problem Challenges in learning good hyperbolic representations using Euclidean operations.
method Proposes HKConv, a trainable hyperbolic convolution that correlates local features with kernel points and aggregates them.
result HKConv learns expressive local features according to hyperbolic geometry and enjoys equivariance to permutation and invariance to parallel transport.
Harmonic maps from hyperbolic planes to hyperbolic space exist with given boundary data.
problem Existence of harmonic maps from product of hyperbolic planes to hyperbolic space.
method Existence result for asymptotic Dirichlet problem.
result Existence of harmonic maps with given boundary data.
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…
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.
A symplectic form is called hyperbolic if its pull-back to the universal cover is a differential of a bounded one-form. The present paper is concerned with the properties and constructions of manifolds admitting hyperbolic symplectic forms. The main results are: * If a symplectic form represents a bounded cohomology cl…
Study shows certain 4D hyperbolic links don't contain geodesic 3-manifolds.
problem Proving certain hyperbolic link complements don't contain geodesic 3-manifolds.
method Analyzing hyperbolic link complements of 2-tori in S^4.
result Proves certain hyperbolic link complements do not contain closed embedded totally geodesic hyperbolic 3-manifolds.
Crochet models of a hyperbolic plane is a popular educational tool as they help to visualize complicated objets in hyperbolic geometry. We present another way how to make crochet models when we view them as a part of a triangulated hyperbolic plane. We also provide a model of a cylinder in a hyperbolic space. This appr…
Study on hyperbolic manifolds with special boundaries.
problem Characterizing geometric properties of hyperbolic manifolds with specific boundaries.
method Analyzing asymptotically hyperbolic manifolds with a horospherical boundary.
result Derived geometric formulas for the model case of horoballs and their complements.
Study shows partial hyperbolicity leads to Anosov dynamics in 3-manifolds.
problem Understanding dynamics in hyperbolic 3-manifolds and Seifert manifolds.
method Classification of partially hyperbolic diffeomorphisms and pseudo-Anosov dynamics.
result Complete classification of partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds and Seifert manifolds.
We provide a simple, combinatorial criteria for a hierarchically hyperbolic space to be relatively hyperbolic by proving a new formulation of relative hyperbolicity in terms of hierarchy structures. In the case of clean hierarchically hyperbolic groups, this criteria characterizes relative hyperbolicity. We apply our c…
The study examines whether a specific type of hyperbolic manifolds remains unchanged under birational transformations.
problem Birational invariance of balanced hyperbolic manifolds.
method Analysis of the class of balanced hyperbolic manifolds.
result The class of balanced hyperbolic manifolds is birationally invariant.
Study connects Kähler and non-Kähler hyperbolicity.
problem Understanding hyperbolicity in Kähler and non-Kähler settings.
method Establishes a connection between SKT and balanced hyperbolicity.
result Highlights relationship with Kähler hyperbolicity.
In this note, we study deformations of quaternionic hyperbolic lattices in larger quaternionic hyperbolic spaces and prove local rigidity results. On the other hand, surface groups are shown to be more flexible in quaternionic hyperbolic plane than in complex hyperbolic plane.
Proves Farrell--Jones Conjecture for hyperbolic groups and their automorphisms.
problem Proving the Farrell--Jones Conjecture for automorphisms of hyperbolic groups.
method Analyzes JSJ decompositions and applies results to automorphisms of hyperbolic groups.
result Proves the fibred Farrell--Jones Conjecture for a class of relatively hyperbolic groups.
The study shows how quotients of mapping class groups are hierarchically hyperbolic.
problem Understanding the hierarchical hyperbolicity of mapping class groups and their quotients.
method A combinatorial criterion for hierarchical hyperbolicity applied to mapping class groups.
result Quotients of mapping class groups by large powers of Dehn twists are hierarchically hyperbolic.
Classifies low-volume hyperbolic 3-manifolds with a maximal cusp.
problem Identifying hyperbolic 3-manifolds with minimal volume and maximal cusps.
method Maximal cusp volume classification and analysis of low-volume manifolds.
result Figure-8 knot complement is unique in certain volume and filling categories.
Introduces new hyperbolicity concepts for complex manifolds.
problem Generalizing hyperbolicity concepts to non-Kähler manifolds.
method Introduces balanced hyperbolicity and divisorial hyperbolicity.
result Every balanced hyperbolic manifold is also divisorially hyperbolic.
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.
The hyperbolic plane admits a quasi-isometric embedding into a hyperbolic group if and only if the group is not virtually free.
The paper proves Gromov hyperbolicity of certain metrics using isoperimetric inequalities.
problem Investigating Gromov hyperbolicity of specific metrics.
method Using isoperimetric inequalities to characterize Gromov hyperbolicity.
result Characterization of domains where these metrics are Gromov hyperbolic.
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…
The study finds a bound on the shortest geodesics in hyperbolic 3-manifolds.
problem Finding bounds on the shortest geodesics in hyperbolic 3-manifolds.
method Establishing an upper bound for the length of the nth shortest closed geodesic in terms of the volume of the manifold. result An upper bound for the length of the nth shortest closed geodesic in terms of the volume of the manifold. Study shows Poisson boundary matches hyperbolic boundary for certain groups.
problem Identifying Poisson boundary for hyperbolic groups without moment conditions.
method Proved using finite entropy random walks and extended to groups with WPD elements.
result Poisson boundary matches hyperbolic boundary for specified groups.
Unified framework for hyperbolic embeddings from mixed data types.
problem Computing hyperbolic embeddings from noisy metric and non-metric data.
method Semidefinite programming and spectral factorization methods.
result Efficient computation of hyperbolic embeddings from arbitrary data.
A new learning method using hyperbolic geometry for class labels.
problem Class label representation and learning in machine learning.
method Hyperbolic Prototype Learning with a new loss function based on hyperbolic geometry.
result Hyperbolic Prototype Learning is equivalent to logistic regression in the one-dimensional case.
Representing data in hyperbolic space can effectively capture latent hierarchical relationships. With the goal of enabling accurate classification of points in hyperbolic space while respecting their hyperbolic geometry, we introduce hyperbolic SVM, a hyperbolic formulation of support vector machine classifiers, and el…
Complex hyperbolic manifolds with many totally geodesic submanifolds are arithmetic.
problem Characterizing arithmeticity of complex hyperbolic manifolds with certain submanifolds.
method Developing superrigidity theorems for complex hyperbolic lattices and proving nonexistence of certain maps.
result Finite volume complex hyperbolic n-manifolds containing infinitely many maximal totally geodesic submanifolds of dimension at least two are arithmetic. 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.
Random quotients preserve hyperbolic properties in groups.
problem Preserving hyperbolic properties in random group quotients.
method Independent random walks, spinning families, projection complexes.
result Random quotients of acylindrical and hierarchical hyperbolic groups remain so.