Let $ρ_n(V)$ be the number of complete hyperbolic manifolds of dimension n with volume less than $V$. Burger, Gelander, Lubotzky, and Moses showed that when n>3 there exist a,b>0 depending on the dimension such that aV log(V) < log(ρ_n(V)) < bV log(V), for V >> 0. In this note, we use their methods to bound the number of hyperbolic manifolds with diameter less than d and show that the number grows double-exponentially. Additionally, this bound holds in dimension 3.
A report on an article of Calegari, Marques and Neves about counting minimal surfaces. An idea of a proof is included using the concept of a laminar measure.
In this article we explore the relationship between the systole and the diameter of closed hyperbolic orientable surfaces. We show that they satisfy a certain inequality, which can be used to deduce that their ratio has a (genus dependent) upper bound.
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 $$N_{M,w}(T) = \sum\limits_{λ_n \leq T}(T-λ_n)^w $$ where $\{λ_n\}$ is the set of eigenvalues of the Laplacian which acts on the space of smooth functions on $M$. If $M$ is non-compact, then the weighted counting function is defined via the inverse Laplace transform. Now let $M_{\ell}$ denote a dege…
We show that the number of isometry classes of cusped hyperbolic $3$-manifolds that bound geometrically grows at least super-exponentially with their volume, both in the arithmetic and non-arithmetic settings.
We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension $2$. We prove a Mertens counting formula for the rational points over a definite quaternion algebra $A$ over $\mathbb Q$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over $A$ in quaternionic Heisenberg groups.
For a conformally compact manifold that is hyperbolic near infinity and of dimension $n+1$, we complete the proof of the optimal $O(r^{n+1})$ upper bound on the resonance counting function, correcting a mistake in the existing literature. In the case of a compactly supported perturbation of a hyperbolic manifold, we establish a Poisson formula expressing the regularized wave trace as a sum over scattering resonances. This leads to an $r^{n+1}$ lower bound on the counting function for scattering poles.
For closed odd-dimensional manifolds with sectional curvature less or equal than -1, we define the minimal surface entropy that counts the number of surface subgroups. It attains the minimum if and only if the metric is hyperbolic. Moreover, we compute the entropy associated with other locally symmetric spaces and cusped hyperbolic 3-manifolds.
We study orbital functions associated to finitely generated geometrically infinite Kleinian groups acting on the hyperbolic space $\mathbb{H}^3$, developing a new method based on the use of the Brownian motion. On the way, we give some estimates of the orbital function associated to nilpotent covers of compact hyperbolic manifolds, partially answering a question asked by M. Pollicott to the author.
Counting surface subgroups in cusped hyperbolic 3-manifolds
Fudan University（复旦大学）；Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；Rutgers University–New Brunswick（罗格斯大学-新布伦瑞克校区）；University of Wisconsin–Madison（威斯康星大学麦迪逊分校）
We strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on complex hyperbolic spaces, especially in dimension $2$. We prove a Mertens' formula for the integer points over a quadratic imaginary number fields $K$ in the light cone of Hermitian forms, as well as an equidistribution theorem of the set of rational points over $K$ in Heisenberg groups. We give a counting formula for the cubic points over $K$ in the complex projective plane whose Galois conjugates are orthogonal…
The aim of this paper is to give an upper bound for the intrinsic diameter of a surface with boundary immersed in a conformally flat three dimensional Riemannian manifold in terms of the integral of the mean curvature and of the length of its boundary. Of particular interest is the application of the inequality to minimal surfaces in the three-sphere and in the hyperbolic space. Here the result implies an a priori estimate for connected solutions of Plateau's problem, as well as a necessary condition on the boundary data for the existence of such solutions. The proof follows a construction of …
Counting Salem numbers arising from arithmetic hyperbolic orbifolds
University of Minnesota（明尼苏达大学）；Universidade Federal Fluminense（弗鲁米嫩塞联邦大学）；Centro de Investigación en Matemáticas AC（数学研究中心AC）；Universiteit Utrecht（乌得勒支大学）
We study the spectral theory of asymptotically hyperbolic manifolds with ends of warped product type. Our main result is an upper bound on the resonance counting function with a geometric constant expressed in terms of the respective Weyl constants for the core of the manifold and the base manifold defining the ends.
We introduced an asymptotic quantity that counts area-minimizing surfaces in negatively curved closed 3-manifolds and show that quantity to only be minimized, among all metrics of sectional curvature less than or equal -1, by the hyperbolic metric.
We prove prime geodesic theorems counting primitive closed geodesics on a compact hyperbolic 3-manifold with length and holonomy in prescribed intervals, which are allowed to shrink. Our results imply effective equidistribution of holonomy and have both the rate of shrinking and the strength of the error term fully symmetric in length and holonomy.
We establish a precise asymptotic formula for the number of homotopy classes of periodic orbits for the geodesic flow on rank one manifolds of nonpositive curvature. This extends a celebrated result of G. A. Margulis to the nonuniformly hyperbolic case and strengthens previous results by G. Knieper. We also establish some useful properties of the measure of maximal entropy.
Suppose that $(X, g)$ is a conformally compact $(n+1)$-dimensional manifold that is hyperbolic at infinity in the sense that outside of a compact set $K \subset X$ the sectional curvatures of $g$ are identically equal to minus one. We prove that the counting function for the resolvent resonances has maximal order of growth $(n+1)$ generically for such manifolds.
We define a norm on homology of punctured tori equipped with a complete hyperbolic metric of finite volume and use it to find asymptotics on the growth of the number of simple geodesics of bounded length.
The maximal analytic Schwarzschild spacetime is manifestly inextendible as a Lorentzian manifold with a twice continuously differentiable metric. In this paper, we prove the stronger statement that it is even inextendible as a Lorentzian manifold with a continuous metric. To capture the obstruction to continuous extensions through the curvature singularity, we introduce the notion of the spacelike diameter of a globally hyperbolic region of a Lorentzian manifold with a merely continuous metric and give a sufficient condition for the spacelike diameter to be finite. The investigation of low-reg…
We establish a sharp geometric constant for the upper bound on the resonance counting function for surfaces with hyperbolic ends. An arbitrary metric is allowed within some compact core, and the ends may be of hyperbolic planar, funnel, or cusp type. The constant in the upper bound depends only on the volume of the core and the length parameters associated to the funnel or hyperbolic planar ends. Our estimate is sharp in that it reproduces the exact asymptotic constant in the case of finite-area surfaces with hyperbolic cusp ends, and also in the case of funnel ends with Dirichlet boundary con…
We show that the length $R$ of a systole of a closed hyperbolic $n$-manifold $(n \geq 3)$ admitting a triangulation by $t$ $n$-simplices can be bounded below by a function of $n$ and $t$, namely \[ R \geq \frac{1}{2^{(nt)^{O(n^4t)} }} .\] We do this by finding a relation between the number of $n$-simplices and the diameter of the manifold and by giving explicit bounds for a well known relation between the length of the core curve of a Margulis tube and its radius. We prove the same result for finite volume manifolds, with a similar but slightly more involved proof.
The purpose of this article is to produce effective versions of some rigidity results in algebra and geometry. On the geometric side, we focus on the spectrum of primitive geodesic lengths (resp., complex lengths) for arithmetic hyperbolic 2-manifolds (resp., 3-manifolds). By work of Reid, this spectrum determines the commensurability class of the 2-manifold (resp., 3-manifold). We establish effective versions of these rigidity results by ensuring that, for two incommensurable arithmetic manifolds of bounded volume, the length sets (resp., the complex length sets) must disagree for a length th…
Let $M_g$ be the moduli space of hyperbolic surfaces of genus $g$ endowed with the Weil-Petersson metric. In this paper, we show that for any $ε>0$, as genus $g$ goes to infinity, a generic surface $X\in M_g$ satisfies that the first eigenvalue $λ_1(X)>\frac{3}{16}-ε$. As an application, we also show that a generic surface $X\in M_g$ satisfies that the diameter $\mathrm{diam}(X)<(4+ε)\ln(g)$ for large genus.
Let $M$ be a compact hyperbolic 3-manifold of diameter $d$ and volume $\leq V$. If $μ_i(M)$ denotes the $i$-th egenvalue of the Hodge laplacian acting on coexact 1-forms of $M$, we prove that $μ_1(M)\geq \frac c{d^3e^{2kd}}$ and $μ_{k+1}(M)\geq \frac c{d^2}$, where $c>0$ depends only on $V$, and $k$ is the number of connected component of the thin part of $M$. Moreover, we prove that for any finite volume hyperbolic 3-manifold $M_\infty$ with cusps, there is a sequence $M_i$ of compact fillings of $M_\infty$ of diameter $d_i\to+\infty$ such that $μ_1(M_i)\geq \frac c{d_i^2}$.
We show that, for horoconvex domains in the hyperbolic space, the product of their fundamental gap with the square of their diameter has no positive lower bound. The result follows from the study of the fundamental gap of geodesic balls as the radius goes to infinity. In the process, we improve the lower bound for the first eigenvalue of balls in hyperbolic space.
In this paper, we numerically investigate the length spectra and the low-lying eigenvalue spectra of the Laplace-Beltrami operator for a large number of small compact(closed) hyperbolic (CH) 3-manifolds. The first non-zero eigenvalues have been successfully computed using the periodic orbit sum method, which are compared with various geometric quantities such as volume, diameter and length of the shortest periodic geodesic of the manifolds. The deviation of low-lying eigenvalue spectra of manifolds converging to a cusped hyperbolic manifold from the asymptotic distribution has been measured by…
