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.
The study finds minimal volume hyperbolic 4-manifolds with embedded 3-manifolds.
problem Existence of small volume hyperbolic 4-manifolds with embedded 3-manifolds.
method Analysis of hyperbolic manifolds and their submanifolds.
result Minimal volume hyperbolic 4-manifolds with embedded 3-manifolds exist.
Algorithm decides if two hyperbolic 3-manifolds are homeomorphic.
problem Determining if two hyperbolic 3-manifolds are homeomorphic.
method Algorithm finds hyperbolic structures and compares geometric manifolds.
result Algorithm runs in bounded time for triangulations with at most t tetrahedra. New examples of hyperbolic Riemannian manifolds proven.
problem Understanding hyperbolic Riemannian manifolds.
method Proving an analogue of the Brody lemma and presenting new examples.
result New examples of Kobayashi hyperbolic Riemannian manifolds.
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.
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 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 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.
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.
Four hyperbolic 24-cell 4-manifolds with one cusp are identified.
problem Identifying hyperbolic 24-cell 4-manifolds with one cusp.
method Analyzing hyperbolic geometry and isometry.
result Found four one-cusped hyperbolic 4-manifolds of minimum volume.
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.
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.
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…
The study explores weakly p-Kähler hyperbolic manifolds.
problem Generalization and application of weakly p-Kähler hyperbolic manifolds. method Investigation of generalizations and applications.
result Exploration of weakly p-Kähler hyperbolic manifolds. This paper is the second in a series whose goal is to understand the structure of low-volume complete orientable hyperbolic 3-manifolds. Using Mom technology, we prove that any one-cusped hyperbolic 3-manifold with volume <= 2.848 can be obtained by a Dehn filling on one of 21 cusped hyperbolic 3-manifolds. We also sho…
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.
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. 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.
Complex hyperbolic triangle groups yield specific 3-manifolds at infinity.
problem Characterizing manifolds at infinity of complex hyperbolic orbifolds.
method Spherical CR uniformization and Dehn surgery.
result Specific 3-manifolds at infinity of complex hyperbolic triangle groups are identified.
Systoles of hyperbolic manifolds are dense and related to Salem numbers.
problem Density of systoles in hyperbolic manifolds.
method Analyzing systoles and arithmetic hyperbolic manifolds.
result Systoles of closed arithmetic hyperbolic manifolds are dense in (0,+∞). 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. Paper computes G-index for specific hyperbolic manifolds.
problem Computing G-index for closed, spin, hyperbolic manifolds.
method Developed techniques to compute G-index for 2- or 4-manifolds.
result Computed G-index for Davis hyperbolic 4-manifold.
The Euler number of special symplectic hyperbolic manifolds is positive.
problem Understanding the Euler number of symplectic hyperbolic manifolds.
method Study L2-harmonic forms on the universal covering space and prove the Singer conjecture. result The Euler number of a special symplectic manifold satisfies (−1)nχ(X)>0. 4 flat 3-manifolds realized in hyperbolic 4-space.
problem Realizing flat 3-manifolds in hyperbolic 4-space.
method Using finite-volume hyperbolic 4-manifolds with cusp sections.
result 6 closed orientable flat 3-manifolds realized as cusp sections.
Estimates intersection pairing in hyperbolic 4-manifolds.
problem Estimating intersection pairing in hyperbolic 4-manifolds.
method Using Thurston norms of homology classes.
result Proved an estimate on intersection pairing.
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.
Census of 10-tetrahedra hyperbolic 3-manifolds with 150,730 new examples.
problem Census of orientable cusped hyperbolic 3-manifolds up to 10 tetrahedra.
method Complete enumeration of minimal ideal triangulations.
result 439,898 new exceptional Dehn fillings and simplest knot exteriors.
The paper finds lower bounds on hyperbolic 3-manifold volumes.
problem Finding lower bounds on hyperbolic 3-manifold volumes.
method Decomposing a 3-manifold into hyperbolic pieces and summing their volumes.
result The volume of a 3-manifold is bounded below by the sum of the volumes of its hyperbolic pieces.
The study explores geometric properties of hyperbolic cohomology classes on Kähler manifolds.
problem Understanding the geometric effects of hyperbolic cohomology classes on Kähler manifolds.
method Introducing Kähler topologically hyperbolic manifolds and proving spectral gap theorems for positive holomorphic Hermitian vector bundles.
result Kähler topologically hyperbolic manifolds are not uniruled nor bimeromorphic to compact Kähler manifolds with trivial first real Chern class.
We construct a locally hyperbolic 3-manifold M such that π1(M) has no divisible subgroups. We then show that M is not homotopy equivalent to any complete hyperbolic manifold.
A homotopy equivalence between a hyperbolic 3-manifold and a closed irreducible 3-manifold is homotopic to a homeomorphsim provided the hyperbolic manifold satisfies a purely geometric condition. There are no known examples of hyperbolic 3-manifolds which do not satisfy this condition.
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 paper finds hyperbolic small knots in many 3-manifolds.
problem Finding small knots in 3-manifolds.
method Explicit examples of hyperbolic small knots in spherical 3-manifolds.
result Explicit examples of hyperbolic small knots in most spherical 3-manifolds.
3D hyperbolic spaces have endless simple paths.
problem Finding simple paths in complex 3D spaces.
method Analyzing geodesics in hyperbolic 3-manifolds.
result Cusped hyperbolic 3-manifolds have infinitely many simple closed geodesics.
Researchers found the minimum volume of a 3-cusped hyperbolic 3-manifold.
problem Finding the minimum volume of a 3-cusped orientable hyperbolic 3-manifold.
method Using guts in sutured and pared manifolds.
result The volume of a 3-cusped orientable hyperbolic 3-manifold is at least 5.49... = 6 × Catalan's constant.
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.
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. Stability of positive mass theorem for hyperbolic manifolds studied.
problem Stability of the positive mass theorem for asymptotically hyperbolic manifolds.
method Adapted intrinsic flat distance approach to show stability for a class of manifolds.
result Stability of the positive mass theorem for a class of asymptotically hyperbolic graphical manifolds.
The study of systoles in arithmetic hyperbolic manifolds.
problem Understanding the systoles of arithmetic hyperbolic manifolds.
method Construction and analysis of arithmetic hyperbolic manifolds.
result Explicit bounds on volumes and systoles of arithmetic hyperbolic manifolds.
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.
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…
The paper constructs contact-hyperbolic manifolds with large automorphism groups.
problem Finding contact-hyperbolic manifolds with large automorphism groups.
method Holomorphic contact structures, pseudometrics, and symplectic quotients.
result Explicit examples of contact-hyperbolic contact manifolds are constructed.
The hyperbolization process affects the structure of manifolds.
problem Understanding the structure of manifolds under hyperbolization.
method Applying hyperbolization to polyhedra and PL submanifolds, and using Ontaneda's Riemannian hyperbolization.
result Find open complete manifolds with specific curvature properties.
The paper applies combinatorial Ricci flows to prove hyperbolic structures on 3-manifolds.
problem Proving the existence of hyperbolic structures on 3-manifolds with cusps.
method Combinatorial Ricci curvature flow methods to study pseudo 3-manifolds and ideal triangulations.
result The extended Ricci flow converges to a decorated hyperbolic polyhedral metric if and only if there exists a zero Ricci curvature metric.
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.
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 examples of hyperbolic 3-manifolds with unique profinite structure.
problem Finding hyperbolic 3-manifolds with unique profinite structure.
method Examining fundamental groups of closed fibered hyperbolic 3-manifolds.
result First examples of closed fibered hyperbolic 3-manifolds with unique profinite structure.