arXiv research

Let $S$ be a compact hyperbolic Riemann surface of genus $g \geq 2$. We call a systole a shortest simple closed geodesic in $S$ and denote by $\mathop{sys}(S)$ its length. Let $\mathop{msys(g)}$ be the maximal value that $\mathop{sys}(\cdot)$ can attain among the compact Riemann surfaces of genus $g$. We call a (global…

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.

Equivalence shown between two mathematical concepts for hyperbolic surfaces.

The study shows how certain surfaces can be filled by hyperbolic manifolds.

New surfaces defy traditional pants decompositions.

Study intersection numbers, lengths, and shortest geodesics on hyperbolic surfaces.

Upper bounds on area for surfaces with constant mean curvature in hyperbolic 3-manifolds.

Researchers compute trace formula for magnetic Laplacian on hyperbolic surfaces.

Bi-geodesic mappings preserve distances on hyperbolic surfaces with boundaries.

Extending BTZ models to complete hyperbolic surfaces.

Researchers solve a Plateau problem for maximal surfaces in pseudo-hyperbolic spaces.

Compact hyperbolic complex manifolds are rigid under deformation.

Classifies Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.

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…

For compact and for convex co-compact oriented hyperbolic surfaces, we prove an explicit correspondence between classical Ruelle resonant states and quantum resonant states, except at negative integers where the correspondence involves holomorphic sections of line bundles.

Given a Delaunay decomposition of a compact hyperbolic surface, one may record the topological data of the decomposition, together with the intersection angles between the `empty disks' circumscribing the regions of the decomposition. The main result of this paper is a characterization of when a given topological decom…

Harmonic metrics are established for SO0(n,n)-Higgs bundles on non-compact hyperbolic surfaces.

We prove there exists a compact embedded minimal surface in a complete finite volume hyperbolic $3$-manifold $\mathcal{N}$. We also obtain a least area, incompressible, properly embedded, finite topology, $2$-sided surface. We prove a properly embedded minimal surface of bounded curvature has finite topology. This dete…

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 $n \geq 4$. 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…

The paper proves geometric bordisms for specific hyperbolic surfaces.

The entropy of geodesic currents on hyperbolic surfaces is bounded by their self-intersection number.

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…

The study of orthospectrum and simple orthospectrum of hyperbolic surfaces.

The paper contains a new proof that a complete, non-compact hyperbolic $3$-manifold $M$ with finite volume contains an immersed, closed, quasi-Fuchsian surface.

A Fuchsian polyhedron in hyperbolic space is a polyhedral surface invariant under the action of a Fuchsian group of isometries (i.e. a group of isometries leaving globally invariant a totally geodesic surface, on which it acts cocompactly). The induced metric on a convex Fuchsian polyhedron is isometric to a hyperbolic…

We develop an asymptotic expansion of the spectral measures on a degenerating family of hyperbolic Riemann surfaces of finite volume. As an application of our results, we study the asymptotic behavior of weighted counting functions, which, if $M$ is compact, is defined for $w \geq 0$ and $T > 0$ by $$N_{M,w}(T) = \sum\…

We discuss the art and science of producing conformally correct euclidean and hyperbolic tilings of compact surfaces. As an example, we present a tiling of the Chmutov surface by hyperbolic (2, 4, 6) triangles.

Non-compact convex sets in hyperbolic 3-space are rigid under isometries.

For compact Riemann surfaces, the collar theorem and Bers' partition theorem are major tools for working with simple closed geodesics. The main goal of this paper is to prove similar theorems for hyperbolic cone-surfaces. Hyperbolic two-dimensional orbifolds are a particular case of such surfaces. We consider all cone …

We prove that any nonabelian, non-Fuchsian representation of a surface group into PSL(2,R) is the holonomy of a folded hyperbolic structure on the surface. Using similar ideas, we establish that any non-Fuchsian representation rho of a surface group into PSL(2,R) is strictly dominated by some Fuchsian representation j,…

Polynomial bound on surfaces in hyperbolic 3-manifolds.

CMC-1 surfaces found on compact Riemann surfaces with Cantor sets.

Defined a new graph type for compact surfaces, proving its connectedness and infinite diameter.

The paper proves constant mean curvature surfaces in specific manifold types.

Study shows exponential growth of Laplacian determinant on random hyperbolic surfaces.

The paper proves the existence of a unique circle packing on hyperbolic surfaces.

Harmonic metrics on Higgs bundles over non-compact hyperbolic surfaces are studied.

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

New invariant for hyperbolic surfaces, geometric criterion for domains.

We use meromorphic quadratic differentials with higher order poles to parametrize the Teichmüller space of crowned hyperbolic surfaces. Such a surface is obtained on uniformizing a compact Riemann surface with marked points on its boundary components, and has non-compact ends with boundary cusps. This extends Wolf's pa…

We show that for each aspherical compact complex surface $X$ whose fundamental group $π$ fits into a short exact sequence $1\to K \to π\to π_1(S) \to 1$ where $S$ is a compact hyperbolic Riemann surface and the group $K$ is finitely-presentable, there is a complex structure on $S$ and a nonsingular holomorphic fibr…

We exhibit an algorithm to determine the bridge number of a hyperbolic knot in the 3-sphere. The proof uses adaptations of almost normal surface theory for compact surfaces with boundary in ideally triangulated knot exteriors.

The group of conformal diffeomorphisms and the group of causal automorphisms on two-dimensional globally hyperbolic spacetimes are clarified. It is shown that if spacetimes have non-compact Cauchy surfaces, then the groups are subgroups of that of two-dimensional Minkowski spacetime, and if spacetimes have compact Cauc…

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

The study finds that certain hyperbolic manifolds contain subgroups isomorphic to surface groups.

The study finds infinite meridional essential surfaces in hyperbolic knot exteriors of various genera.

We prove a partial generalization of Bonahon's tameness result to surfaces inside irreducible 3-manifolds with hyperbolic fundamental group. Bonahon's result states that geometrically infinite ends of freely indecomposable hyperbolic 3-manifolds are simply degenerate. It is easy to see that a geometrically infinite end…

Improved bound on the product of first Laplacian eigenvalue and area for genus three surfaces.