Related papers: Orthospectrum and simple orthospectrum rigidity: f…

The main goal of this article is to understand how the length spectrum of a random surface depends on its genus. Here a random surface means a surface obtained by randomly gluing together an even number of triangles carrying a fixed metric.…

We consider geometric triangulations of surfaces, i.e., triangulations whose edges can be realized by disjoint locally geodesic segments. We prove that the flip graph of geometric triangulations with fixed vertices of a flat torus or a…

We show that every orientable infinite-type surface is properly rigid as a consequence of a more general result. Namely, we prove that if a homotopy equivalence between any two non-compact orientable surfaces is a proper map, then it is…

Topology and geometry are deeply intertwined in the study of surfaces, though their interaction manifests differently in smooth and discrete settings. In the smooth category, a classical result asserts that any closed smooth surface…

The rigidity of marked length spectrum for closed hyperbolic surfaces due to Fricke-Klein [7] has been the motivation of many different rigidity results, specially for manifolds of negative curvature. From the works of Vigneras [18], Sunada…

We construct examples of flat surfaces in $\mathbb{H}^3$ which are graphs over a two-punctured horosphere and classify complete embedded flat surfaces in $\mathbb{H}^3$ with only one end and at most two isolated singularities.

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

In this article, we study the Dirac spectrum of typical hyperbolic surfaces of finite area, equipped with a nontrivial spin structure (so that the Dirac spectrum is discrete). For random Weil-Petersson surfaces of large genus $g$ with…

This paper presents some finiteness results for the number of boundary slopes of immersed essential surfaces of given genus g in a compact 3-manifold with torus boundary. In the case of hyperbolic 3-manifolds we obtain uniform quadratic…

We explore for compact Riemannian surfaces whose boundary consists of a single closed geodesic the relationship between orthospectrum and boundary length. More precisely, we establish a uniform lower bound on the boundary length in terms of…

On a surface with a Finsler metric, we investigate the asymptotic growth of the number of closed geodesics of length less than $L$ which minimize length among all geodesic multicurves in the same homology class. An important class of…

In this article we study the spectrum of totally geodesic surfaces of a finite volume hyperbolic 3-manifold. We show that for arithmetic hyperbolic 3-manifolds that contain a totally geodesic surface, this spectrum determines the…

Let $Z$ be a projective hypersurface such that its underlying reduced variety has only isolated singularities. In case its irreducible components have constant multiplicities, for instance if $\dim Z>1$, we show that the spectrum of its…

We study the geometry of homogeneous hypersurfaces and their focal sets in complex hyperbolic spaces. In particular, we provide a characterization of the focal set in terms of its second fundamental form and determine the principal…

We prove that curve complexes of surfaces are finitely rigid: for every orientable surface S of finite topological type, we identify a finite subcomplex X of the curve complex C(S) such that every locally injective simplicial map from X…

We prove that oriented and standard shadowing properties are equivalent for topological flows on closed surfaces with the nonwandering set consisting of the finite number of critical elements (i.e., singularities or closed orbits).…

For hyperbolic surfaces with geodesic boundary, we study the orthosystole, i.e. the length of a shortest essential arc from the boundary to the boundary. We recover and extend work by Bavard completely characterizing the surfaces maximizing…

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,…

We describe the structure of the singular sets of constant curvature, convex hypersurfaces in hyperbolic space for general convex curvature functions. We apply this result to the study of the ideal Plateau problem in hyperbolic space for…

The goal of this work is to give new quantitative results about the distribution of semi-arithmetic hyperbolic surfaces in the moduli space of closed hyperbolic surfaces. We show that two coverings of genus $g$ of a fixed arithmetic surface…