Shortest non-simple closed geodesics on hyperbolic surfaces found.
problem Finding the shortest non-simple closed geodesics on hyperbolic surfaces.
method Analyzing closed geodesics with at least k self-intersections on hyperbolic surfaces.
result The shortest non-simple closed geodesics lie on an ideal pair of pants and have length $2\arccosh(2k+1)$.
First example of a hyperbolic 4-orbifold underlying P2.
problem Finding closed hyperbolic 4-orbifolds with symplectic underlying spaces.
method Realized P2 as the underlying space of a closed hyperbolic 4-orbifold. result First example of a closed hyperbolic 4-orbifold with symplectic underlying space.
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.
Upper bounds on geodesics in hyperbolic manifolds.
problem Bounding the number of shortest closed geodesics in hyperbolic manifolds.
method Using the Selberg trace formula, we derive upper bounds on the number of shortest and primitive geodesics.
result Generalized Parlier's theorem to higher dimensions and uniform bounds for manifolds with bounded geometry.
The paper studies the shortest closed multi-geodesics on hyperbolic surfaces as their genus grows.
problem Finding the asymptotic behavior of shortest closed multi-geodesics on hyperbolic surfaces.
method Analyzing the length of shortest filling closed multi-geodesics using hyperbolic geometry and asymptotic analysis.
result The length of a shortest filling closed multi-geodesic is uniformly comparable to a specific formula involving the genus and lengths of closed geodesics.
We prove the existence of Alexandrov embedded closed magnetic geodesics on closed hyperbolic surfaces. Closed magnetic geodesics correspond to closed curves with prescribed geodesic curvature.
Upper bound established for the length of shortest closed geodesics in hyperbolic link complements.
problem Finding bounds on the length of shortest closed geodesics in hyperbolic link complements.
method Established an upper bound for the length of an nth shortest closed geodesic as a logarithmic function of the volume of the manifold.
result An upper bound of the length of an nth shortest closed geodesic is established as a logarithmic function of the volume of the manifold.
Minimal geodesics on hyperbolic surfaces are long.
problem Finding the shortest closed geodesics on hyperbolic surfaces.
method Analyzing the self-intersection number to estimate geodesic lengths.
result The minimal length of geodesics grows logarithmically with the self-intersection number.
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.
The paper studies hyperbolic phenomena on closed surfaces using bicorn curves.
problem Understanding hyperbolic phenomena on curve graphs of closed surfaces.
method Using the theory of bicorn curves to analyze the curve graphs of closed surfaces.
result Proves that the curve graph of any closed surface is 15-hyperbolic with one exception.
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…
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,+∞). Develops a method to define and characterize geodesics on hyperbolic surfaces.
problem Characterizing closed geodesics on hyperbolic surfaces without self-intersection.
method Constructive definition of the Goldman bracket using closed geodesics.
result Algebraic characterization of geodesics on hyperbolic surfaces.
This paper contains examples of closed aspherical manifolds obtained as a by-product of recent work by the author [arXiv:math.GR/0509490] on the relative strict hyperbolization of polyhedra. The following is proved. (I) Any closed aspherical triangulated n-manifold M^n with hyperbolic fundamental group is a retract of …
We prove Calegari's conjecture that every quasigeodesic flow on a closed hyperbolic 3-manifold has closed orbits.
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.
Constructs hyperbolic surfaces with small eigenvalues.
problem Finding hyperbolic surfaces with eigenvalues below a given threshold.
method Geometric proof using techniques from B.Randol's 1974 paper.
result Constructs closed hyperbolic covering surfaces with eigenvalues less than any small positive number ε.
No geometric minimal foliations in hyperbolic 3-manifolds.
problem Existence of geometric minimal foliations in hyperbolic three-manifolds.
method Analyzing three-dimensional closed hyperbolic manifolds.
result No locally geometric 1-parameter family of closed minimal surfaces.
Study on embedding hyperbolic 2-orbifolds in Bianchi orbifolds.
problem Embedding closed totally geodesic hyperbolic 2-orbifolds in Bianchi orbifolds.
method Analyzing Bianchi orbifolds H3/PSL(2,Od) for large d. result Existence of at least cd closed embedded totally geodesic hyperbolic 2-orbifolds for large d. Random subsurfaces of hyperbolic surfaces equidistribute to ribbon graphs.
problem Distribution of shapes of complementary subsurfaces in moduli space.
method Study of shapes of complementary subsurfaces in moduli space as boundary lengths go to infinity.
result Random subsurfaces look like random ribbon graphs.
Closed hyperbolic manifolds and manifolds with nonpositive sectional curvature are geometrically rigid under certain curvature conditions.
problem Geometric rigidity under scalar curvature lower bound
method Prove rigidity in the equality case of the sharp bottom spectrum estimate
result Closed manifolds with specific curvature conditions must be hyperbolic
Sharp bound on smallest diameter of hyperbolic surfaces.
problem Finding the smallest possible diameter of hyperbolic surfaces.
method Proved a specific formula for the minimal diameter.
result Minimal diameter is log(g)+25loglog(g)+O(1). We show that every hyperbolic knot complement contains a closed quasi-Fuchsian surface.
Motivated by classical theorems on minimal surface theory in compact hyperbolic three-manifolds, we investigate the questions of existence and deformations for least area minimal surfaces in complete noncompact hyperbolic three-manifold of finite volume. We prove any closed immersed incompressible surface can be deform…
In this paper, we will use Kahn-Markovic's almost totally geodesic surfaces to construct certain π1-injective 2-complexes in closed hyperbolic 3-manifolds. Such 2-complexes are locally almost totally geodesic except along a 1-dimensional subcomplex. Using Agol and Wise's result that fundamental groups of hyperbolic …
Among other things, we prove the following two topologcal statements about closed hyperbolic 3-manifolds. First, every rational second homology class of a closed hyperbolic 3-manifold has a positve integral multiple represented by an oriented connected closed π1-injectively immersed quasi-Fuchsian subsurface. Second…
Geodesics on hyperbolic surfaces become evenly spread over time.
problem Distribution of geodesics on hyperbolic surfaces.
method Equidistribution analysis of geodesics with length less than T as T approaches infinity.
result Closed geodesics on hyperbolic surfaces of finite area become evenly spread over time.
The entropy of minimal surfaces is minimized in hyperbolic manifolds.
problem Counting essential minimal surfaces in closed negatively curved manifolds.
method Defining minimal surface entropy and computing it for various spaces.
result Minimal surface entropy is minimized in hyperbolic manifolds.
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. Clarifies definitions of global hyperbolicity in various spaces.
problem Clarifying terminology in recent literature on global hyperbolicity.
method Comparing definitions in Lorentzian length spaces, optimal transport, and topological preordered spaces.
result The causal relation is a closed order and preserves compactness in all cases.
We define and study hyperbolic extensions.
We describe a class of genus 2 closed hyperbolic 3-manifolds of arbitrarily large volume.
We correct and complete a conjecture of D. Gabai, R. Meyerhoff and N. Thurston on the classification and properties of thin tubed closed hyperbolic 3-manifolds. We additionally show that if N is a closed hyperbolic 3-manifold, then either N=Vol3 or N contains a closed geodesic that is the core of an embedded tube of ra…
New conditions found for hyperbolic bicycle tracks.
problem Conditions for hyperbolic bicycle tracks.
method Hyperbolic development interpretation of bicycling monodromy.
result New necessary and sufficient conditions for hyperbolic bicycle tracks.
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.
We prove that for any \e>0, there exists a closed hyperbolic 4-manifold with a closed geodesic of length < \e.
This paper determines which orientable hyperbolic 3-manifolds contain simple closed geodesics. The Fuchsian group corresponding to the thrice-punctured sphere generates the only example of a complete non-elementary orientable hyperbolic 3-manifold that does not contain a simple closed geodesic. We do not assume that th…
The paper proves rigidity of length identities for simple closed curves on hyperbolic surfaces.
problem Characterizing hyperbolic surfaces by their simple length spectra.
method Proving rigidity of length identities over Teichmüller spaces.
result Simple length spectra can be used as moduli for generic hyperbolic surfaces.
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.
The study of geodesics on hyperbolic surfaces with angle and side bounds.
problem Understanding the geometric properties of geodesics on hyperbolic surfaces.
method Analyzing the angles and sides of polygons formed by geodesics, based on their lengths.
result Bounds for angles and sides of polygons formed by geodesics, depending only on geodesic lengths.
We prove that every closed oriented 3-manifold admits a hyperbolic cone-manifold structure with cone-angle arbitrarily close to 2pi.
We define and study "hyperbolic forcing".
Study of closed trajectories in hyperbolic plane with specific curvature constraints.
problem Critical trajectories in hyperbolic plane for a specific energy function.
method Classification of critical trajectories based on momentum causal character, proof of existence of closed trajectories.
result Existence of countably many closed trajectories with time-like momentum.
We investigate the maximal solid tubes around short simple geodesics in hyperbolic three-manifolds and how complex length of curves relate to closed, incompressible, least area minimal surfaces. As applications, we prove, there are some closed hyperbolic three-manifolds fibering over the circle which are not foliated b…
The study finds hyperbolic manifolds without spin^c structures in dimensions 5 and above.
problem Existence of closed hyperbolic manifolds without spin^c structures.
method Proof of existence and commensurability classes for manifolds with non-vanishing third Stiefel-Whitney class.
result Infinitely many commensurability classes of closed hyperbolic manifolds without spin^c structures.
Closed finite gap curves are dense in Sobolev spaces of curves in hyperbolic and Euclidean 3-spaces.
problem Density of closed finite gap curves in Sobolev spaces.
method Proof of density in Sobolev spaces for curves in hyperbolic and Euclidean 3-spaces.
result Closed finite gap curves are dense in Sobolev spaces of curves in hyperbolic and Euclidean 3-spaces.
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.
Study on shortest geodesics crossing multiple times on hyperbolic surfaces with cusps.
problem Finding shortest geodesics crossing multiple times on hyperbolic surfaces.
method Investigates closed geodesics on hyperbolic surfaces with at least one cusp, focusing on minimal length and self-intersection numbers.
result For large enough k, self-intersection numbers are exactly k for geodesics crossing at least k times.