Related papers: Non-simple systoles on random hyperbolic surfaces …

We prove that all hierarchically hyperbolic spaces have finite asymptotic dimension and obtain strong bounds on these dimensions. One application of this result is to obtain the sharpest known bound on the asymptotic dimension of the…

A closed hyperbolic surface of genus $g\ge 2$ can be decomposed into pairs of pants along shortest closed geodesics and if these curves are sufficiently short (and with lengths uniformly bounded away from 0), then the geometry of the…

This article introduces a universal moduli space for the set whose archetypal element is a pair that consists of a metric and second fundamental form from a compact, oriented, positive genus minimal surface in some hyperbolic 3-manifold.…

We calculate the asymptotic average rate at which a generic geodesic on a finite area hyperbolic 2-orbifold returns to an embedded disc on the surface, as well as the average amount of time it spends in the disc during each visit. This…

We study deformations of complex hyperbolic surfaces which furnish the simplest examples of: (i) negatively curved K\"ahler manifolds and (ii) negatively curved Riemannian manifolds not having {\it constant} curvature. Although such complex…

Conformally compact asymptotically hyperbolic metrics have been intensively studied. The goal of this note is to understand what intrinsic conditions on a complete Riemannian manifold (M,g) will ensure that g is asymptotically hyperbolic in…

In this paper we study the systoles of arithmetic hyperbolic 2- and 3-manifolds. Our first result is the construction of infinitely many arithmetic hyperbolic 2- and 3-manifolds which are pairwise noncommensurable, all have the same…

We show that for every $n\geq 2$ and any $\epsilon>0$ there exists a compact hyperbolic $n$-manifold with a closed geodesic of length less than $\epsilon$. When $\epsilon$ is sufficiently small these manifolds are non-arithmetic, and they…

Our goal is to show, in two different contexts, that "random" surfaces have large pants decompositions. First we show that there are hyperbolic surfaces of genus $g$ for which any pants decomposition requires curves of total length at least…

We consider the geodesic flow on a complete connected negatively curved manifold. We show that the set of invariant borel probability measures contains a dense $G_\delta$-subset consisting of ergodic measures fully supported on the…

The first non-zero Laplace eigenvalue of a hyperbolic surface, or its spectral gap, measures how well-connected the surface is: surfaces with a large spectral gap are hard to cut in pieces, have a small diameter and fast mixing times. For…

We apply topological methods to study the smallest non-zero number $\lambda_1$ in the spectrum of the Laplacian on finite area hyperbolic surfaces. For closed hyperbolic surfaces of genus two we show that the set $\{S \in {\mathcal{M}_2}:…

In this paper, we first construct a sequence of hyperbolic surfaces with connected geodesic boundary such that the first normalized Steklov eigenvalue $\tilde{\sigma}_1$ tends to infinity. We then prove that as $g\rightarrow \infty$, a…

A holomorphic curve in moduli spaces is the image of a non-constant holomorphic map from a hyperbolic surface $B$ of type $(g,n)$ to the moduli space $\mathcal{M}_h$ of closed Riemann surfaces of genus $h$. We show that, when all peripheral…

We present a construction of sequences of closed hyperbolic surfaces that have long systoles which form pants decompositions of these surfaces. The length of the systoles of these surfaces grows logarithmically as a function of their genus.

We study the asymptotics of the Weil-Petersson volumes of the moduli spaces of compact Riemann surfaces of genus $g$ with $n$ punctures, for fixed $n$ as $g \to \infty$.

We prove that if $\Sigma$ is a closed surface of genus at least 3 and $G$ is a split real semisimple Lie group of rank at least $3$ acting faithfully by isometries on a symmetric space $N$, then there exists a Hitchin representation…

This note is about a type of quantitative density of closed geodesics on closed hyperbolic surfaces. The main results are upper bounds on the length of the shortest closed geodesic that $\varepsilon$-fills the surface.

We consider the ortho spectrum of hyperbolic surfaces with totally geodesic boundary. We show that in general the ortho spectrum does not determine the systolic length but that there are only finitely many possibilities. As a corollary we…

We prove a universal inequality between the diastole, defined using a minimax process on the one-cycle space, and the area of closed Riemannian surfaces. Roughly speaking, we show that any closed Riemannian surface can be swept out by a…