Compact manifolds with specific cover properties are hyperbolic.
problem Understanding Gromov hyperbolicity in compact manifolds.
method Proving Gromov hyperbolicity through coboundary expansion in residual covers.
result Compact manifolds with certain cover properties have hyperbolic fundamental groups.
The study connects hyperbolicity and isoperimetric inequality in manifolds and graphs.
problem Understanding the relationship between hyperbolicity and isoperimetric inequality.
method Characterization of hyperbolic manifolds and graphs with isoperimetric inequality.
result Characterization of trees with isoperimetric inequality and solvability of Dirichlet problem at infinity.
Develop criteria to distinguish Gromov-Thurston manifolds using algebraic Dehn fillings.
problem Distinguishing homotopy types of Gromov-Thurston manifolds
method Using virtual Dehn fillings of relatively hyperbolic groups
result Criteria to distinguish homotopy types
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.
Potential theory extended to Gromov hyperbolic spaces.
problem Extending potential theory to a new class of spaces.
method Unified framework for Gromov hyperbolic metric measure spaces.
result Boundary Harnack inequalities and complete classification of positive harmonic functions.
The paper characterizes hyperbolic manifolds and graphs verifying a specific isoperimetric inequality.
problem Understanding the relationship between hyperbolicity and isoperimetric inequalities in manifolds and graphs.
method Characterization of hyperbolic manifolds and graphs with isoperimetric inequality, using Gromov boundary.
result Having a pole is a necessary condition for verifying the isoperimetric inequality, which can be removed.
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.
Harmonic maps from curved spaces to hyperbolic ones exist and are Lipschitz.
problem Existence and properties of harmonic maps between curved and hyperbolic spaces.
method Energy minimization and finite distance analysis.
result Harmonic maps from Hadamard manifolds to Gromov hyperbolic mCAT(0)-spaces are Lipschitz. Gromov-Thurston covers have Betti numbers as expected.
problem Understanding Betti numbers of branched covers of hyperbolic manifolds.
method Analyzing Gromov-Thurston branched covers and their Betti numbers.
result Betti numbers match expectations for non-divisible degree covers.
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.
The study constructs AdS manifolds from Gromov-Thurston manifolds.
problem Creating hyperbolic and anti-de Sitter structures from Gromov-Thurston manifolds.
method Explicit correspondence between quasifuchsian AdS manifolds and compact quotients of Ø(2d,2)/U(d,1).
result Existence of quasifuchsian AdS manifolds and hyperbolic ends with specified boundary.
The paper embeds non-arithmetic hyperbolic manifolds into higher-dimensional spaces.
problem Embedding non-arithmetic hyperbolic manifolds into higher-dimensional hyperbolic spaces.
method Using totally geodesic submanifolds and commensurability classes.
result Many non-arithmetic hyperbolic manifolds can be embedded geodesically.
New trick builds hyperbolic manifolds from compact ones, proving some don't virtually fiber.
problem Proving some hyperbolic manifolds don't virtually fiber.
method Hyperbolic reflection group trick, embedding theory, manifold topology.
result Constructed Gromov hyperbolic 7-manifolds that don't virtually fiber over a circle.
New proof for Ending Lamination Theorem using hyperbolic 3-simplices.
problem Proving the Ending Lamination Theorem for hyperbolic 3-manifolds of infinite volume.
method Relies on Maximum Volume Law for hyperbolic 3-simplices.
result New proof of Ending Lamination Theorem.
Classifies 3D partially hyperbolic systems, proving ergodicity.
problem Ergodicity of partially hyperbolic diffeomorphisms in 3-manifolds.
method Topological classification, Anosov flows, foliations, Gromov hyperbolicity.
result Complete answer to Hertz-Hertz-Ures conjecture for 3D systems.
The paper finds many thin subgroups isomorphic to Gromov-Piatetski-Shapiro lattices.
problem Understanding thin subgroups in special linear groups.
method Constructing and embedding non-arithmetic hyperbolic manifolds into SL(n+1)(R).
result Non-arithmetic lattices in SO(n,1) can be embedded into SL(n+1)(R) as thin subgroups.
Minimal action on universal circle for foliations on 3-manifolds.
problem Action of fundamental group on universal circle of foliations.
method Analyzes uniform foliations on 3-manifolds, proving minimality and transitivity.
result Action is minimal and transitive on pairs of different points.
Simplified potential theory for complex geometric spaces.
problem Developing potential theory for Gromov hyperbolic manifolds.
method Streamlining arguments to use coarse geometro-analytic invariants.
result Derived useful details not found in original sources.
Study shows rigidity of polyhedrons in hyperbolic spaces.
problem Rigidity of polyhedrons in hyperbolic spaces.
method Extending Gromov's comparison theory to metrics with negative scalar curvature lower bounds.
result Localization of the positive mass theorem for asymptotically hyperbolic manifolds.
The study finds the minimum average area ratio on hyperbolic manifolds and its relation to scalar curvature.
problem Finding the minimum average area ratio on hyperbolic manifolds.
method Analyzing the average area ratio and normalized total scalar curvature for hyperbolic n-manifolds.
result The average area ratio attains a local minimum of 1 at the hyperbolic metric.
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.
We show that trees of manifolds, the topological spaces introduced by Jakobsche, appear as boundaries at infinity of various spaces and groups. In particular, they appear as Gromov boundaries of some hyperbolic groups, of arbitrary dimension, obtained by the procedure of strict hyperbolization. We also recognize these …
We present new lower bounds on the complexity of Dehn surgery manifolds of knots, using our recent result on the Cheeger-Gromov rho invariants and triangulations. As an application, we give explicit examples of closed hyperbolic 3-manifolds with fixed first homology for which the gap between the Gromov norm and the com…
New 4D hyperbolic spaces found with minimal volume.
problem Finding minimal-volume hyperbolic 4-manifolds.
method Applying Gromov and Piatetski-Shapiro's technique to construct non-arithmetic manifolds.
result Proved existence of at least 2 commensurability classes of minimal-volume hyperbolic 4-manifolds, and constructed the smallest known non-arithmetic one.
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.
New graph-based complexity measure for hyperbolic 3-manifolds.
problem Defining a complexity measure for hyperbolic 3-manifolds.
method Using Morse functions to graphs and Gromov area.
result Linear relationship between graph adjacency and metric complexity.
New examples of hyperbolic manifolds with convex projective structures.
problem Finding new examples of hyperbolic manifolds with convex projective structures.
method Generalized Dehn filling and real projective structures.
result Infinitely many hyperbolic manifolds with cusps admit properly convex real projective structures.
This paper extends boundary embedding results to coarsely convex spaces.
problem Generalizing boundary embedding results to coarsely convex spaces.
method Generalizing Dydak and Virk's work on Gromov hyperbolic spaces to coarsely convex spaces.
result Maps between coarsely convex spaces induce continuous maps between their boundaries.
Estimates simplicial volume for complex hyperbolic surfaces.
problem Bounding the simplicial volume of complex hyperbolic surfaces.
method Estimates Gromov norm and uses top dimensional class in Hc4. result Explicit upper bound for simplicial volume.
We show that for a generic simple closed curve C in the asymptotic boundary of a Gromov hyperbolic 3-space with cocompact metric X, there exist a unique least area plane P in X with asymptotic boundary C. This result has interesting topological applications for constructions of canonical 2-dimensional objects in 3-mani…
Polyhedra rigidity theorem in hyperbolic space proved.
problem Dihedral rigidity conjecture in hyperbolic 3-space.
method Comparison theorem for polyhedra in a 3-manifold with scalar curvature bounded below.
result Confirms Gromov dihedral rigidity conjecture in hyperbolic 3-space.
The study proves Gromov hyperbolicity for certain complex domains.
problem Characterizing Gromov hyperbolicity for complex domains.
method Analyzing domains in C2 with finite d'Angelo type and using automorphisms. result Domains in C2 with finite d'Angelo type are Gromov hyperbolic. New real hyperbolic manifolds constructed over surfaces with maximal Euler number ratio.
problem Constructing real hyperbolic manifolds with specific topological properties.
method Applying techniques from [AGG] and using the Gromov-Lawson-Thurston conjecture [GLT].
result New examples achieve the maximal value of $|eM/χS|= rac35$.
Classifies mapping tori of specific groups, generalizing known results.
problem Classifying mapping tori of specific groups.
method Using Hopf-type properties and Poincaré Duality groups.
result Generalizes and provides new proofs for fibered 3-manifolds.
Study on non-Gromov hyperbolic tube domains and their geometric properties.
problem Characterizing non-Gromov hyperbolic tube domains with convex bases.
method Provided a criterion for non-Gromov hyperbolicity, studied Hilbert metric, and continuity properties of complex geodesics.
result Similarity of geometry of tube domains and convex domains, connections between metrics.
The paper examines hyperbolicity in bounded strongly minimally convex domains in R^d.
problem Investigating hyperbolicity in bounded strongly minimally convex domains.
method Analyzing the minimal metric and Hilbert metric in convex domains.
result Every bounded strongly minimally convex domain is Gromov hyperbolic.
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.
New finding links hyperbolic manifold systolic volume to triangulation complexity.
problem Understanding the relationship between systolic volume and triangulation complexity in hyperbolic manifolds.
method Proof based on Jørgensen and Thurston's theorem of hyperbolic volume.
result Systolic volume of hyperbolic manifolds is related to triangulation complexity.
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.
Metric transforms make Euclidean half lines hyperbolic, preserving geodesic properties.
problem Characterizing metric transforms that make Euclidean half lines hyperbolic.
method Characterization of metric transforms φ such that ([0,∞),∣⋅∣φ) is Gromov hyperbolic. result Metric transform rigidity for roughly geodesic Gromov hyperbolic spaces.
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.
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…
The paper connects Turaev-Viro invariants and Gromov norm of 3-manifolds.
problem Relating Turaev-Viro invariants and Gromov norm of 3-manifolds.
method Combines TQFT techniques, geometric decomposition theory, and analytical estimates of 6j-symbols. result Established a relation between the asymptotics of Turaev-Viro invariants and Gromov norm of 3-manifolds.
New findings on leafwise quasi-geodesic foliations in 3-manifolds.
problem Understanding leafwise quasi-geodesic foliations in 3-manifolds.
method Analyzing intersections of transverse foliations in 3-manifolds with Gromov hyperbolic leaves.
result Hausdorff leafspace condition for leafwise quasi-geodesic foliations.
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.
We give a necessary complex geometric condition for a bounded smooth convex domain in Cn, endowed with the Kobayashi distance, to be Gromov hyperbolic. More precisely, we prove that if a smooth bounded convex domain contains an analytic disk in its boundary, then the domain is not Gromov hyperbolic for the Kobayashi di…
3-manifold groups have free subgroups in certain covers.
problem Finding free subgroups in 3-manifold groups.
method Shows existence of a tower of covers with specific properties.
result Subgroups generated by a certain number of elements are free.
The study shows almost maximal volume entropy rigidity for certain manifolds with integral Ricci curvature.
problem Volume entropy rigidity for manifolds with lower integral Ricci curvature bound.
method Analyzing manifolds with specific integral Ricci curvature bounds, diameter, and volume entropy.
result The universal cover of the manifold is close to a hyperbolic space form under certain conditions.