The study classifies all compact hyperbolic polytopes with eight facets.
problem Classifying compact hyperbolic Coxeter four-polytopes with specific numbers of facets.
method Complete classification through mathematical analysis.
result The complete classification of compact hyperbolic Coxeter four-polytopes with eight facets.
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.
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 classifies all compact 5D polytopes with 9 facets.
problem Classifying compact hyperbolic Coxeter polytopes.
method Complete classification through mathematical analysis.
result A complete list of compact hyperbolic Coxeter 5D polytopes with 9 facets.
Proves hyperbolized groups are virtually compact special and linear.
problem Proving hyperbolized groups are virtually compact special and linear.
method Constructing an action of a hyperbolized group on a dual CAT(0) cubical complex.
result Proves hyperbolized groups are virtually compact special and linear.
The purpose of the present paper is to prove existence of super-exponentially many compact orientable hyperbolic arithmetic n-manifolds that are geometric boundaries of compact orientable hyperbolic (n+1)-manifolds, for any n≥2, thereby establishing that these classes of manifolds have the same growth rate w…
Compact hyperbolic complex manifolds are rigid under deformation.
problem Studying the deformation behavior of compact hyperbolic complex manifolds.
method Analyzing smooth families of compact complex manifolds over the unit disk and compact Riemann surfaces.
result The H-locus is either at most a discrete subset or the whole domain, depending on the family structure. In this paper we establish a strong multiplicity one type property for the length-holonomy spectrum for the three dimensional compact hyperbolic spaces. We use the analytic properties of Selberg-Gangolli-Wakayama zeta functions associated to compact hyperbolic spaces.
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 paper classifies compact hyperbolic Coxeter polytopes and improves upper bounds.
problem Classifying compact hyperbolic Coxeter polytopes and understanding their combinatorial properties.
method Study of imes0-products of Lannér diagrams, proving superhyperbolic properties, and analyzing Lannér subdiagrams. result Improved upper bounds on the dimension of compact hyperbolic Coxeter polytopes.
Unique hyperbolic manifolds identified by boundary pleating.
problem Identifying hyperbolic manifolds from their boundary pleating.
method Used pleating measured lamination on the boundary of convex cores.
result Convex co-compact hyperbolic manifolds are uniquely determined by their pleating lamination.
Geometric constraints help classify hyperbolic polytopes.
problem Classifying reflective anisotropic Lorentzian lattices and cocompact arithmetic hyperbolic reflection groups.
method Established geometric constraints on compact Coxeter polytopes in hyperbolic spaces.
result Geometric constraints are useful for classifying hyperbolic polytopes.
The paper characterizes convex co-compact groups with one-dimensional boundary faces.
problem Characterizing convex co-compact groups with specific boundary properties.
method Proving relative hyperbolicity and using coarse Hilbert dimension.
result Convex co-compact groups with one-dimensional boundary faces are relatively hyperbolic.
Conditions for trivial gradient hyperbolic Ricci and Yamabe solitons to be Einstein or constant scalar curvature.
problem Characterizing conditions for gradient hyperbolic Ricci and Yamabe solitons to be trivial.
method Analyzing Lie derivatives and divergence conditions.
result Conditions for compact gradient hyperbolic Yamabe solitons to be trivial, leading to constant scalar curvature.
We prove a strong multiplicity one theorem for the length spectrum of compact even dimensional hyperbolic spaces i.e. if all but finitely many closed geodesics for two compact even dimensional hyperbolic spaces have the same length, then all closed geodesics have the same length.
Wise's Quasiconvex Hierarchy Theorem classifying hyperbolic virtually compact special groups in terms of quasiconvex hierarchies played an essential role in Agol's proof of the Virtual Haken Conjecture. Answering a question of Wise, we construct a new virtual quasiconvex hierarchy for relatively hyperbolic virtually co…
Study properties of balanced hyperbolic compact complex manifolds.
problem Understanding cohomology and harmonic spaces of balanced hyperbolic manifolds.
method Proved vanishing theorems and Hard Lefschetz-type theorems for balanced hyperbolic compact complex manifolds.
result Non-existence of certain L1 currents on the universal covering space of a balanced hyperbolic manifold. We demonstrate how to construct three-dimensional compact hyperbolic polyhedra using Newton's Method. Under the restriction that the dihedral angles are non-obtuse, Andreev's Theorem provides as necessary and sufficient conditions five classes of linear inequalities for the dihedral angles of a compact hyperbolic polyh…
Equivalence shown between two mathematical concepts for hyperbolic surfaces.
problem None explicitly stated, but related to mathematical equivalence of concepts.
method Benjamini-Schramm convergence and zeta functions equivalence demonstration.
result Equivalence of Benjamini-Schramm convergence and zeta functions for compact hyperbolic surfaces.
We determine the three hyperbolic 5-orbifolds of smallest volume among compact arithmetic orbifolds, and we identify their fundamental groups with hyperbolic Coxeter groups. This gives two different ways to compute the volume of these orbifolds.
For k>6, we determine the minimal area of a compact hyperbolic surface, and an oriented compact hyperbolic surface that can be tiled by embedded regular triangles of angle 2π/k. Based on this, all the cases of equality in Laszlo Fejes Toth's triangle bound for hyperbolic surfaces are described.
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.
Kerckhoff and Storm conjectured that compact hyperbolic n-orbifolds with totally geodesic boundary are infinitesimally rigid when n>3. This paper verifies this conjecture for a specific example based on the 4-dimensional hyperbolic 120-cell.
We study the existence of starshaped compact hypersurfaces with prescribed m-th mean curvature in hyperbolic space.
We show that any compact orientable hyperbolic 3-cone-manifold with cone angle at most πcan be continuously deformed to a complete hyperbolic manifold homeomorphic to the complement of the singularity. This together with the local rigidity by Hodgson and Kerckhoff implies the global rigidity for compact orientable hype…
Paper defines dynamical coherence for flows and proves it under specific conditions.
problem Understanding the dynamics of partially hyperbolic flows.
method Introduces dynamical coherence and proves it for flows with a specific foliation.
result Dynamical coherence proved for flows with a particular foliation.
The study shows a finite number of groups acting on hyperbolic spaces with bounded entropy and compact quotient.
problem Finite number of groups acting on hyperbolic spaces with bounded entropy and compact quotient.
method Analyzing torsion-free groups acting by isometries on hyperbolic metric spaces with bounded entropy and compact quotient.
result The set of such groups is finite and can be estimated based on hyperbolicity constant, entropy, and quotient diameter.
We prove that for n>4 there is no compact arithmetic hyperbolic n-manifold whose Euler characteristic has absolute value equal to 2. In particular, this shows the nonexistence of arithmetically defined hyperbolic rational homology n-sphere with n even different than 4.
We consider a hyperbolic Dirac-type operator with growing potential on a a spatially non-compact globally hyperbolic manifold. We show that the Atiyah-Patodi-Singer boundary value problem for such operator is Fredholm and obtain a formula for this index in terms of the local integrals and the relative eta-invariant int…
The study shows how certain surfaces can be filled by hyperbolic manifolds.
problem Conditions for a Riemann surface to be filled by hyperbolic manifolds.
method Analyzes conditions involving short hyperbolic curves on the surface.
result Riemann surfaces can be filled by hyperbolic manifolds of negative volume.
Study on Cheeger constant in specific hyperbolic 3-manifolds.
problem Analyzing Cheeger constant in convex co-compact hyperbolic 3-manifolds.
method Examining Cheeger constant as a functional in the space of specific hyperbolic 3-manifolds.
result Global maximum of Cheeger constant is uniquely attained at the Fuchsian locus.
Classifies Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.
problem Identifying Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.
method Comprehensive classification of commensurability classes of cusped, arithmetic, and non-arithmetic complex hyperbolic 2-manifolds.
result Some Nil 3-manifolds are cross-sections in every commensurability class, while others are cross-sections in only one.
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.
Let S be a compact hyperbolic Riemann surface of genus g≥2. We call a systole a shortest simple closed geodesic in S and denote by sys(S) its length. Let msys(g) be the maximal value that sys(⋅) can attain among the compact Riemann surfaces of genus g. We call a (global…
Paper proves stability of positive mass theorem for specific types of manifolds.
problem Stability of positive mass theorem for compact graphical manifolds.
method Used Federer--Fleming flat distance and static quasi-local Brown-York energy.
result Proved stability of positive mass theorem for compact (locally) hyperbolic graphical manifolds.
We show that for compact orientable hyperbolic orbisurfaces, the Laplace spectrum determines the length spectrum as well as the number of singular points of a given order. The converse also holds, giving a full generalization of Huber's theorem to the setting of compact orientable hyperbolic orbisurfaces.
Generalizes global hyperbolicity to higher signatures and proves compactness.
problem Global hyperbolicity in higher pseudo-Riemannian signatures.
method Generalization and proof of compactness of causal diamonds.
result Existence of solutions to a Plateau problem and characterization of holonomies.
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…
New hyperbolic 3-manifolds have many neighbors.
problem Finding hyperbolic 3-manifolds with a large number of neighbors.
method Constructed a sequence of non-compact finite volume hyperbolic 3-manifolds.
result The kissing number grows at least as fast as the volume to the power of 31/27 - ε.
Given a harmonic measure of a hyperbolic lamination on a compact metric space, a positive harmonic function is defined on the universal cover of a typical leaves. We discuss some properties of this function. Especially if all the leaves are hyperbolic, ergodic harmonic measures are divided into two classes.
The paper builds complex hyperbolic 2-manifolds with isolated singularities.
problem Finding compact complex hyperbolic 2-manifolds with non-free actions and isolated fixed points.
method Constructs specific examples for each prime p with Z/pZ action. result General examples for p=2 related to complex hyperbolic lattices conjugacy separability. 3-manifold groups can only have convex co-compact representations if they are geometric or hyperbolic.
problem Understanding which 3-manifold groups can have convex co-compact representations.
method Analyzing representations of 3-manifold groups into projective general linear group, focusing on convex co-compactness.
result Fundamental groups of closed irreducible orientable 3-manifolds can only admit convex co-compact representations if they are geometric or hyperbolic.
The classical Brody's theorem asserts the equivalence between two notions of hyperbolicity for compact complex spaces, one named after Kobayashi and one expressed in terms of lack of non constant holomorphic entire functions (compactness is only used to prove the harder implication). We extend this theorem to Deligne-M…
We show that the definition of global hyperbolicity in terms of the compactness of the causal diamonds and non-total imprisonment can be extended to spacetimes with continuous metrics, while retaining all of the equivalences to other notions of global hyperbolicity. In fact, global hyperbolicity is equivalent to the co…
Extending BTZ models to complete hyperbolic surfaces.
problem Extending BTZ models to complete hyperbolic surfaces.
method Proving a parametrization result for globally hyperbolic Cauchy-maximal and Cauchy-compact locally Minkowski manifolds with extreme BTZ.
result The tangent bundle of the Teichmüller space parametrizes globally hyperbolic Cauchy-maximal and Cauchy-compact locally Minkowski manifolds with extreme BTZ.
We exhibit the first examples of compact orientable hyperbolic manifolds that do not have any spin structure. We show that such manifolds exist in all dimensions n≥4. The core of the argument is the construction of a compact orientable hyperbolic 4-manifold M that contains a surface S of genus 3 with sel…
Study intersection numbers, lengths, and shortest geodesics on hyperbolic surfaces.
problem Understanding the relationship between intersection numbers, lengths, and shortest geodesics on hyperbolic surfaces.
method Analyzing the asymptotic behavior of interaction strength I(X) as X approaches infinity in the moduli space of compact hyperbolic surfaces.
result Determined the asymptotic behavior of interaction strength I(X) in terms of the length of the shortest geodesic sys(X).
Paper proves embedding theorem for conformally compact manifolds.
problem Embedding conformally compact manifolds into hyperbolic spaces.
method Proves analogous Nash Embedding Theorem for conformally compact manifolds.
result Conformally compact manifolds can be isometrically embedded into hyperbolic spaces.