相关论文: Construction of hyperbolic Riemann surfaces with l…
In this paper, we investigate the asymptotic behavior of the non-simple systole, which is the length of a shortest non-simple closed geodesic, on a random closed hyperbolic surface on the moduli space $\mathcal{M}_g$ of Riemann surfaces of…
Let $(M,g)$ be a closed, oriented, Riemannian manifold of dimension $m$. We call a systole a shortest non-contractible loop in $(M,g)$ and denote by $sys(M,g)$ its length. Let $SR(M,g)=\frac{{sys(M,g)}^m}{vol(M,g)}$ be the systolic ratio of…
The main goal of this note is to show that the study of closed hyperbolic surfaces with maximum length systole is in fact the study of surfaces with maximum length homological systole. The same result is shown to be true for once-punctured…
For any $\varepsilon>0$, we construct a closed hyperbolic surface of genus $g=g(\varepsilon)$ with a set of at most $\varepsilon g$ systoles that fill, meaning that each component of the complement of their union is contractible. This…
We apply a study of orders in quaternion algebras, to the differential geometry of Riemann surfaces. The least length of a closed geodesic on a hyperbolic surface is called its systole, and denoted syspi_1. P. Buser and P. Sarnak…
The systoles of a hyperbolic surface {\Sigma} are the shortest closed geodesics. We say that the systoles fill the surface if the set Syst({\Sigma}) of all systoles cuts {\Sigma} into polygons. We refine an idea of Schmutz [15] to construct…
In this paper, we investigate basic geometric quantities of a random hyperbolic surface of genus $g$ with respect to the Weil-Petersson measure on the moduli space $\mathcal{M}_g$. We show that as $g$ goes to infinity, a generic surface…
The interaction strength I(X) of a compact hyperbolic surface X is the best upper bound for the intersection number of two closed geodesics divided by the product of their lengths. Let $M_g$ be the moduli space of compact hyperbolic…
We examine the large systole problem, which concerns compact hyperbolic Riemannian surfaces whose systole, the length of the shortest noncontractible loops, grows logarithmically in genus. The generalization of a construction of Buser and…
In this thesis we consider a way to construct a rich family of compact Riemann Surfaces in a combinatorial way. Given a 3-regualr graph with orientation, we construct a finite-area hyperbolic Riemann surface by gluing triangles according to…
The systole of a hyperbolic surface is bounded by a logarithmic function of its genus. This bound is sharp, in that there exist sequences of surfaces with genera tending to infinity that attain logarithmically large systoles. These are…
In this article, we provide bounds on systoles associated to a holomorphic $1$-form $\omega$ on a Riemann surface $X$. In particular, we show that if $X$ has genus two, then, up to homotopy, there are at most $10$ systolic loops on…
In this article, we study geometric aspects of semi-arithmetic Riemann surfaces by means of number theory and hyperbolic geometry. First, we show the existence of infinitely many semi-arithmetic Riemann surfaces of various shapes and prove…
We prove Poisson approximation results for the bottom part of the length spectrum of a random closed hyperbolic surface of large genus. Here, a random hyperbolic surface is a surface picked at random using the Weil-Petersson volume form on…
In this paper we examine the geometry of minimal surfaces of arithmetic hyperbolic 3-manifolds. In particular, we give bounds on the totally geodesic 2-systole, construct infinitely many incommensurable manifolds with the same initial…
A basic feature of Teichm\"uller theory of Riemann surfaces is the interplay of two dimensional hyperbolic geometry, the behavior of geodesic-length functions and Weil-Petersson geometry. Let $\mathcal{T}_g$ $(g\geq 2)$ be the Teichm\"uller…
In this paper, we investigate the asymptotics of shortest filling closed multi-geodesics of closed hyperbolic surfaces as systole $\to 0$ or as genus $\to \infty$. We first show that for a closed hyperbolic surface $X_g$ of genus $g$, the…
We obtain the exact values of the systoles of these hyperbolic surfaces of genus $g$ with cyclic symmetries of the maximum order and the next maximum order. Precisely: for genus $g$ hyperbolic surface with order $4g+2$ cyclic symmetry, the…
Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so-called fat graph, which we call the systolic graph. We study which fat graphs are systolic graphs for some surface…
The purpose this article is to try to understand the mysterious coincidence between the asymptotic behavior of the volumes of the Moduli Space of closed hyperbolic surfaces of genus $g$ with respect to the Weil-Petersson metric and the…