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.
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.
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. 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.
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.
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. 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.
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.
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.
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.
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.
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.
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.
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.
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…
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.
We will show that, for any noncompact arithmetic hyperbolic m-manifold with m>3, and any compact arithmetic hyperbolic m-manifold with m>4 that is not a 7-dimensional arithmetic hyperbolic manifold defined by octonions, its fundamental group is not LERF. The main ingredient in the proof is a study on abelia…
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…
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…
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.
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.
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.
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.
Compactify complex hyperbolic almost Hermitian manifolds.
problem Understanding the geometric structure of complex hyperbolic almost Hermitian manifolds.
method Analyzing the asymptotic curvature and boundary conditions of the manifold.
result The interior of a compact almost complex manifold can represent the original manifold.
Thurston's hyperbolization theorem for Haken manifolds and normal surface theory yield an algorithm to determine whether or not a compact orientable 3-manifold with nonempty boundary consisting of tori admits a complete finite-volume hyperbolic metric on its interior. A conjecture of Gabai, Meyerhoff, and Milley reduce…
New cutoff phenomenon found for geodesic paths on hyperbolic manifolds.
problem Understanding the cutoff phenomenon for geodesic paths on hyperbolic manifolds.
method Spectral strategy and detailed spectral analysis of the spherical mean operator.
result Geodesic paths on compact hyperbolic manifolds exhibit cutoff for spatially localized initial conditions.
Study shows infimum of dual volume equals convex core volume for hyperbolic 3-manifolds.
problem Infimum of dual volume of convex co-compact hyperbolic 3-manifolds.
method Varying geometry by quasi-isometric deformations to deduce infimum.
result Linear lower bound on quasi-Fuchsian manifold volume based on bending lamination length.
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 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. 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.
3D hyperbolic manifolds map one-to-one to their boundary character varieties.
problem Mapping 3D hyperbolic manifolds to their boundary character varieties.
method Using Bonahon-Schläfli formula and volume rigidity of discrete co-compact representations.
result Peripheral map is a birational isomorphism with its image.
Paper finds minimum volume for specific anti-de Sitter 3-manifolds.
problem Finding the minimum volume of globally hyperbolic anti-de Sitter 3-manifolds.
method Analyzes maximal globally hyperbolic Cauchy-compact anti-de Sitter 3-manifolds.
result Minimum volume is π²|χ(M)|, attained for Fuchsian manifolds.
Unique compact Fuchsian manifolds with convex boundary are determined by their boundary.
problem Identifying compact Fuchsian manifolds with convex boundaries.
method Proving uniqueness based on the induced path metric on the boundary.
result Compact Fuchsian manifolds with convex boundaries are uniquely determined by the induced path metric on the boundary.
Proves unique hyperbolic metric for 3-manifolds with ideal triangulation.
problem Proving a unique hyperbolic metric for 3-manifolds with specific triangulations.
method Combining combinatorial Ricci flow with ideal triangulation for pseudo 3-manifolds.
result Extended Ricci flow converges to the hyperbolic metric exponentially fast.
We show that the aspherical manifolds produced via the relative strict hyperbolization of polyhedra enjoy many group-theoretic and topological properties of open finite volume negatively pinched manifolds, including relative hyperbolicity, nonvanishing of simplicial volume, co-Hopf property, finiteness of outer automor…
The study reveals a persistent bias in the distribution of holonomy on compact hyperbolic 3-manifolds.
problem The distribution of holonomy on compact hyperbolic 3-manifolds is not uniformly distributed.
method An asymptotic count of closed geodesics by their length and holonomy, and analysis of spectral parameters.
result A normalized, smoothed bias count of holonomy is distributed according to a probability distribution, controlled by the number of zero spectral parameters.
Extends Fried's result to arbitrary representations of compact hyperbolic manifolds.
problem Behavior of dynamical zeta functions at the origin for compact hyperbolic manifolds.
method Uses complex-valued torsion instead of Ray-Singer analytic torsion.
result Holomorphicity and value at s=0 for twisted Ruelle zeta function for arbitrary representations.
We study the inverse resonance problem for conformally compact manifolds which are hyperbolic outside a compact set. Our results include compactness of isoresonant metrics in dimension two and of isophasal negatively curved metrics in dimension three. In dimensions four or higher we prove topological finiteness theorem…
Formula derived for ALH manifolds, showing existence of specific 3D manifolds.
problem Energy calculation and existence of specific 3D conformally compact ALH manifolds.
method Derivation of energy formula and application to manifold existence.
result Existence of 3D conformally compact ALH manifolds with specific properties.
Researchers decompose hyperbolic n-manifolds with totally geodesic boundaries into polyhedral cells.
problem Decomposing hyperbolic n-manifolds with totally geodesic boundaries into polyhedral cells.
method Two different approaches to demonstrate the existence of polyhedral decompositions.
result The number of polyhedral decompositions of M is finite. The isometry group of a compact n-dimensional hyperbolic manifold is known to be finite. We show that for every n > 2, every finite group is realized as the full isometry group of some compact hyperbolic n-manifold. The cases n = 2 and n = 3 have been proven by Greenberg and Kojima, respectively. Our proof is non const…
We present a simple algorithm for finding eigenmodes of the Laplacian for arbitrary compact hyperbolic 3-manifolds. We apply our algorithm to a sample of twelve manifolds and generate a list of the lowest eigenvalues. We also display a selection of eigenmodes taken from the Weeks space.
We derive a new renormalized volume formula for conformally compact asymptotically hyperbolic manifolds in dimension four. The formula generalizes the ones given by Anderson, Albin, and Chang-Qing-Yang for the case of Poincare-Einstein manifolds. We also derive variational formulas for the renormalized seen as a functi…
New hyperbolic manifolds found with same trace ring.
problem Finding non-commensurable hyperbolic manifolds with identical trace rings.
method Proved existence of infinitely many non-commensurable manifolds with same ambient group and trace ring.
result Infinitely many non-commensurable hyperbolic manifolds with the same ambient group and trace ring.
The renormalized volume is reinterpreted using isoperimetric profiles.
problem Understanding the renormalized volume of convex co-compact hyperbolic 3-manifolds.
method Using isoperimetric profiles and Minkowski inequalities.
result A sharp Minkowski inequality for horospherically convex sets in H3. Upper bounds on area for surfaces with constant mean curvature in hyperbolic 3-manifolds.
problem Finding area constraints for surfaces with constant mean curvature in hyperbolic 3-manifolds.
method Established an upper bound on the area of closed embedded surfaces with constant mean curvature at least one, depending on the mean curvature and genus bounds.
result Area bound implies compactness for such surfaces, with specific proportional bounds for Bryant surfaces.
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.