arXiv research

Shortest non-simple closed geodesics on hyperbolic surfaces found.

The paper studies hyperbolic phenomena on closed surfaces using bicorn curves.

Minimal geodesics on hyperbolic surfaces are long.

The paper studies the shortest closed multi-geodesics on hyperbolic surfaces as their genus grows.

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…

Sharp bound on smallest diameter of hyperbolic surfaces.

Develops a method to define and characterize geodesics on hyperbolic surfaces.

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.

Constructs hyperbolic surfaces with small eigenvalues.

Geodesics on hyperbolic surfaces become evenly spread over time.

The entropy of minimal surfaces is minimized in hyperbolic manifolds.

Paper proves existence of minimal surfaces in hyperbolic 3-manifolds.

The paper proves rigidity of length identities for simple closed curves on hyperbolic surfaces.

Geodesic surfaces embed into hyperbolic 3-manifolds for all finite group actions.

The paper proves unique geodesics on hyperbolic surfaces and finds lower bounds.

We show that every hyperbolic knot complement contains a closed quasi-Fuchsian surface.

Plumbing of surfaces embeds in hyperbolic 4-manifolds.

Study on embedding surfaces into 3-manifolds, focusing on equivariant cases.

The paper studies the growth of closed geodesics on hyperbolic surface amalgams.

Random subsurfaces of hyperbolic surfaces equidistribute to ribbon graphs.

Improved bounds on geodesic intersections on hyperbolic surfaces.

Inverts operator on hyperbolic surfaces, constructing invariant distributions.

Unstable minimal surfaces in n-space link to hyperbolic products.

Lower bounds on geodesic length with few intersections on hyperbolic surfaces.

In this article, we prove that the commensurability class of a closed, orientable, hyperbolic 3-manifold is determined by the surface subgroups of its fundamental group. Moreover, we prove that there can be only finitely many closed, orientable, hyperbolic 3-manifolds that have the same set of surfaces.

The study of second eigenvalues of hyperbolic surfaces improves bounds and investigates their behavior for large genus.

A discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hyperbolic polyhedral metric with a given discrete curvature satisfying …

The paper bounds eigenvalue multiplicities for hyperbolic surfaces using short geodesics.

The paper derives upper bounds on eigenvalues of Laplace-Beltrami operator on hyperbolic surfaces.

Study shows spectral gaps limit points on surfaces.

In this note, we consider the rigidity of the focal decomposition of closed hyperbolic surfaces. We show that, generically, the focal decomposition of a closed hyperbolic surface does not allow for non-trivial topological deformations, without changing the hyperbolic structure of the surface. By classical rigidity theo…

The paper shows nearly-Fuchsian properties for certain hyperbolic 3-manifolds.

We show that every hyperbolic link complement contains closed quasi-Fuchsian surfaces. As a consequence, we obtain the result that on a hyperbolic link complement, if we remove from each cusp of the manifold a certain finite set of slopes, then all remaining Dehn fillings on the link complement yield manifolds with clo…

The systole size affects the smallest eigenvalue of hyperbolic surfaces.

New upper bound for Cheeger constant of hyperbolic surfaces.

Let M be a closed hyperbolic 3-manifold. We show that the number of genus g surface subgroups of the fundamental group of M grows like g^{2g}.

Random hyperbolic surfaces are mostly tangle-free, with geometric implications.

Partial coverings of hyperbolic surfaces equidistribute with geodesics.

Currents on cusped hyperbolic surfaces have a denseness property similar to compact surfaces.

The length of shortest non-simple geodesics grows logarithmically with surface genus.

We apply topological methods to study eigenvalues of the Laplacian on closed hyperbolic surfaces. For any closed hyperbolic surface $S$ of genus $g$, we get a geometric lower bound on ${λ_{2g-2}}(S)$: ${λ_{2g-2}}(S) > 1/4 + {ε_0}(S)$, where ${ε_0}(S) > 0$ is an explicit constant which depends only on the systole of $S$

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…

We study compact hyperbolic surface laminations. These are a generalization of closed hyperbolic surfaces which appear to be more suited to the study of Teichmüller theory than arbitrary non-compact surfaces. We show that the Teichmüller space of any non-trivial hyperbolic surface lamination is infinite dimensional. In…

Proved contractibility of geodesic triangulation space on hyperbolic surfaces.

We count the number of conjugacy classes of maximal, genus g, surface subroups in hyperbolic 3-manifold groups. For any closed hyperbolic 3-manifold, we show that there is an upper bound on this number which grows factorially with g. We also give a class of closed hyperbolic 3-manifolds for which there is a lower bound…

For large genus, spectral gaps on hyperbolic surfaces approach a limit.

We give an identity involving sums of functions of lengths of simple closed geodesics, known as a McShane identity, on any non-orientable hyperbolic surface with boundary which generalises Mirzakhani's identities on orientable hyperbolic surfaces with boundary.

We study closed geodesics on hyperbolic surfaces, and give bounds for their angles of intersection and self-intersection, and for the sides of the polygons that they form, depending only on the lengths of the geodesics