Colin Adams, Joel Hass, Peter Scott
This paper determines which orientable hyperbolic 3-manifolds contain simple closed geodesics. The Fuchsian group corresponding to the thrice-punctured sphere generates the only example of a complete non-elementary orientable hyperbolic 3-manifold that does not contain a simple closed geodesic. We do not assume that the manifold is geometrically finite or that it has finitely generated fundamental group.
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This article explores closed geodesics on hyperbolic surfaces. We show that, for sufficiently large $k$, the shortest closed geodesics with at least $k$ self-intersections, taken among all hyperbolic surfaces, all lie on an ideal pair of pants and have length $2\arccosh(2k+1)$.
Let W be a compact hyperbolic n-manifold with totally geodesic boundary. We prove that if n>3 then the holonomy representation of pi_1 (W) into the isometry group of hyperbolic n-space is infinitesimally rigid.
We give an identity involving sums of functions of lengths of simple closed geodesics, known as a McShane identity, on any non-orientable hyperbolic surface with boundary which generalises Mirzakhani's identities on orientable hyperbolic surfaces with boundary.
Geodesic simplices of pseudo-hyperbolic space
本文通过上同调解释伪双曲空间测地单纯形，并给出了其体积有限的充要条件，推导出二维双曲空间中理想测地多面体具有有限体积。
Given a geometrically finite hyperbolic cone-manifold, with the cone singularity sufficiently short, we construct a one parameter family of cone-manifolds decreasing the cone angle to zero. We also control the geometry of this one parameter family via the Schwarzian derivative of the projective boundary and the length of closed geodesics.
This article deals with the set of closed geodesics on complete finite type hyperbolic surfaces. For any non-negative integer $k$, we consider the set of closed geodesics that self-intersect at least $k$ times, and investigate those of minimal length. The main result is that, if the surface has at least one cusp, their self-intersection numbers are exactly $k$ for large enough $k$.
The goal of the article is to provide different explicit quantifications of the non density of simple closed geodesics on hyperbolic surfaces. In particular, we show that within any embedded metric disk on a surface, lies a disk of radius only depending on the topology of the surface (and the size of the first embedded disk), which is disjoint from any simple closed geodesic.
We prove that every complete finite-volume hyperbolic 3-manifold $M$ that is tessellated into (embedded) right-angled regular polyhedra (dodecahedra or ideal octahedra) embeds geodesically in a complete finite-volume connected orientable hyperbolic 4-manifold $W$, which is also tessellated into right-angled regular polytopes (120-cells and ideal 24-cells). If $M$ is connected, then Vol($W$) < $2^{49}$Vol($M$). This applies for instance to the Borromean link complement. As a consequence, the Borromean link complement bounds geometrically a hyperbolic 4-manifold.
Suppose n>2, let M,M' be n-dimensional connected complete finite-volume hyperbolic manifolds with non-empty geodesic boundary, and suppose that the fundamental group of M is quasi-isometric to the fundamental group of M' (with respect to the word metric). Also suppose that if n=3, then the boundaries of M and of M' are compact. We show that M is commensurable with M'. Moreover, we show that there exist homotopically equivalent hyperbolic 3-manifolds with non-compact geodesic boundary which are not commensurable with each other. We also prove that if M is as above and G is a finitely generated …
The polyhedral decomposition of cusped hyperbolic $n$-manifolds with totally geodesic boundary
该研究针对带全测地边界的体积有限非紧完备双曲$n$-流形，证明其存在由理想多面体或带单个截断面的部分截断多面体构成的多面体分解，用两种方法验证结论并证明分解数量有限，完善了相关流形分解理论。
Shortest k-Geodesics on Hyperbolic Surfaces
本文研究双曲面上闭测地线的长度与自交数的关系，改进了现有估计，将最大自交数的渐进行界从512降至128。
$L^2$-bounds for drilling short geodesics in convex co-compact hyperbolic 3-manifolds
本文研究了凸余紧双曲3-流形钻孔短测地线后边界复射影结构的变化，给出了不依赖于边界单射半径的$L^2$-界，证明其变化受普适常数乘以测地线长度平方根的约束。
Let M be a complete finite-volume hyperbolic 3-manifold with compact non-empty geodesic boundary and k toric cusps, and let T be a geometric partially truncated triangulation of M. We show that the variety of solutions of consistency equations for T is a smooth manifold or real dimension 2k near the point representing the unique complete structure on M. As a consequence, the relation between deformations of triangulations and deformations of representations is completely understood, at least in a neighbourhood of the complete structure. This allows us to prove, for example, that small deformat…
We show that cusped finite-volume hyperbolic 3-manifolds contain infinitely many simple closed geodesics.
In this paper we analyze and classify the totally geodesic subspaces of finite volume quaternionic hyperbolic orbifolds and their generalizations, locally symmetric orbifolds arising from irreducible lattices in Lie groups of the form $(\mathbf{Sp}_{2n}(\mathbb{R}))^q \times \prod_{i=1}^r \mathbf{Sp}(p_i,n-p_i) \times (\mathbf{Sp}_{2n}(\mathbb{C}))^s$. We give criteria for when the totally geodesic subspaces of such an orbifold determine its commensurability class. We give a parametrization of the commensurability classes of finite volume quaternionic hyperbolic orbifolds in terms of arithmeti…
We discuss two generalizations of the collar lemma. The first is the stable neighborhood theorem which says that a (not necessarily simple) closed geodesic in a hyperbolic surface has a \lq\lq stable neighborhood\rq\rq whose width only depends on the length of the geodesic. As an application, we show that there is a lower bound for the length of a closed geodesic having crossing number $k$ on a hyperbolic surface. This lower bound tends to infinity with $k$. Our second generalization is to totally geodesic hypersurfaces of hyperbolic manifolds. Namely, we construct a tubular neighborhood funct…
Totally geodesic hyperbolic 3-manifolds in hyperbolic link complements of tori in $S^4$
该研究针对四维球面中二维环面的双曲链环补空间，证明其内部不存在闭的嵌入全测地双曲3-流形，为高维双曲流形的几何结构研究提供了新结论。
The lengths of geodesics on hyperbolic surfaces satisfy intriguing equations, known as identities, relating these lengths to geometric quantities of the surface. This paper is about a large family of identities that relate lengths of closed geodesics and orthogeodesics to boundary lengths or number of cusps. These include, as particular cases, identities due to Basmajian, to McShane and to Mirzakhani and Tan-Wong-Zhang. In stark contrast to previous identities, the identities presented here include the lengths taken among all closed geodesics.
The first examples of totally geodesic Seifert surfaces are constructed for hyperbolic knots and links, including both free and totally knotted surfaces. Then it is proved that two bridge knot complements cannot contain totally geodesic orientable surfaces.
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
Asymptotically geodesic hypersurfaces and the fundamental groups of hyperbolic manifolds
研究双曲流形中渐近测地超曲面的性质，证明其基本群虚特殊且整数线性，进一步构造渐近测地且强填充的超曲面，部分回答Al Assal-Lowe的问题。
Let n>2 and let M be an orientable complete finite volume hyperbolic n-manifold with (possibly empty) geodesic boundary having Riemannian volume vol(M) and simplicial volume ||M||. A celebrated result by Gromov and Thurston states that if M has empty boundary then the ratio between vol(M) and ||M|| is equal to v_n, where v_n is the volume of the regular ideal geodesic n-simplex in hyperbolic n-space. On the contrary, Jungreis and Kuessner proved that if the boundary of M is non-empty, then such a ratio is strictly less than v_n. We prove here that for every a>0 there exists k>0 (only depending…
Combinatorial p-th Calabi Flows for Total Geodesic Curvatures in hyperbolic background geometry
本文研究双曲背景几何下带边界曲面上具锥奇点的广义圆填充，利用变分原理证明其存在性与刚性，并首次引入组合p次Calabi流以求解预设全测地曲率与离散高斯曲率的圆填充。
In the present paper, we show that the minimal length of closed geodesics on finite-type hyperbolic surfaces with self-intersection number $k$ has order $2\log k$ as $k$ gets large.
This thesis mainly treats two developments of the classical theory of hypersurfaces inside pseudo-Riemannian space forms. The former - a joint work with Francesco Bonsante - consists in the study of immersions of smooth manifolds into holomorphic Riemannian space forms of constant curvature -1 (including SL(2,C) with a multiple of its Killing form): this leads to a Gauss-Codazzi theorem, it suggests an approach to holomorphic transitioning of immersions into pseudo-Riemannian space forms, a trick to construct holomorphic maps into the PSL(2,C)-character variety, and leads to a restatement of B…
Asymptotics of shortest filling closed multi-geodesics
本文研究了双曲面中闭合多测地线的渐进行为，证明了其长度与亏格和测地线长度的对数相关，并展示了在随机双曲面模型中的应用。
Let $M$ be a compact hyperbolic $3$-manifold with volume $V$. Let $L$ be a link such that $M\setminus L$ is hyperbolic. For any hyperbolic link $L$ in $M$, in this article, we try to establish an upper bound of the length of $n^{th}$ shortest closed geodesic of $M\setminus L$ in terms of $V$.
Kerckhoff and Storm conjectured that compact hyperbolic n-orbifolds with totally geodesic boundary are infinitesimally rigid when n>3. This paper verifies this conjecture for a specific example based on the 4-dimensional hyperbolic 120-cell.
This notes explores angle structures on ideally triangulated compact $3$-manifolds with high genus boundary. We show that the existence of angle structures implies the existence of a hyperbolic metric with totally geodesic boundary, and conversely each hyperbolic $3$-manifold with totally geodesic boundary has an ideal triangulation that admits angle structures.
Shortest filling geodesics on hyperbolic surfaces
该研究确定了模空间中亏格$g$双曲曲面上填充（多）测地线的最短长度，证明其由补集为直角正$(8g-4)$边形的测地线实现，并构造了相应单条测地线。
Let $M$ be a compact hyperbolic $3$-manifold with volume $V$. Let $L$ be a link such that $M\setminus L$ is hyperbolic. For any hyperbolic link $L$ in $M$, in this article, we establish an upper bound of the length of an $n^{th}$ shortest closed geodesic as a logarithmic function of $V$ in $M\setminus L$. Our works complement the work of Lakeland and Leininger \cite{Christopher} on the upper bound of systole length.
We consider the existence of simple closed geodesics or "geodesic knots" in finite volume orientable hyperbolic 3-manifolds. Previous results show that at least one geodesic knot always exists [Bull. London Math. Soc. 31(1) (1999) 81-86], and that certain arithmetic manifolds contain infinitely many geodesic knots [J. Diff. Geom. 38 (1993) 545-558], [Experimental Mathematics 10(3) (2001) 419-436]. In this paper we show that all cusped orientable finite volume hyperbolic 3-manifolds contain infinitely many geodesic knots. Our proof is constructive, and the infinite family of geodesic knots prod…
Complex hyperbolic surfaces whose closed geodesics are all simple
本文证明了第二类算术复双曲曲面中所有闭测地线均为简单闭测地线，涵盖已知的伪射影平面实例。
On geodesic embeddings of hyperbolic surfaces into hyperbolic 3-manifolds
本文研究闭可定向双曲曲面全测地嵌入闭可定向双曲3-流形的问题，证明所有有限不可约群作用下的准柏拉图曲面均存在等变测地嵌入，并讨论更一般有限群作用的情形。
It is a longstanding problem to determine the precise relationship between the geodesic length spectrum of a hyperbolic manifold and its commensurability class. A well known result of Reid, for instance, shows that the geodesic length spectrum of an arithmetic hyperbolic surface determines the surface's commensurability class. It is known, however, that non-commensurable arithmetic hyperbolic surfaces may share arbitrarily large portions of their length spectra. In this paper we investigate this phenomenon and prove a number of quantitative results about the maximum cardinality of a family of …
Bi-geodesic mappings between hyperbolic surfaces with boundary
该研究证明带边界紧双曲曲面间满足自身与逆映射都保测地的双射必为等距映射，完善了双曲曲面测地映射的相关理论。
Explicit Lie bracket of closed geodesics on a hyperbolic surface with applications
本文开发了一套非欧几里得平面几何方法工具箱，基于闭测地线构造性地定义了有向闭曲线形变类上的Goldman括号，并以此代数刻画双曲曲面上无自相交的闭测地线。
In this paper we examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using a variety of techniques from analytic number theory, we address the following problems: Is the commensurability class of an arithmetic hyperbolic 3-orbifold determined by the lengths of closed geodesics lying on totally geodesic surfaces?, Do there exist arithmetic hyperbolic 3-orbifolds whose "s…
Entropy and self-intersection number of geodesic currents on compact hyperbolic surfaces
研究测地电流在双曲面上的熵与自交数关系，建立熵与自交数及系统最小长度的定量上界，证明小自交数导致小熵。
For each $g \ge 2$, we prove existence of a computable constant $ε(g) > 0$ such that if $S$ is a strongly irreducible Heegaard surface of genus $g$ in a complete hyperbolic 3-manifold $M$ and $γ$ is a simple geodesic of length less than $ε(g)$ in $M$, then $γ$ is isotopic into $S$.
Closed geodesics on hyperbolic surfaces with few intersections
本研究针对完备有限型双曲曲面上自交点数≥2的闭测地线，证明其长度有紧下界$2\log(5+2\sqrt6)$，且该下界可在三穿孔球面的螺旋测地线上取到。
Simple geodesics in closed hyperbolic manifolds
该研究针对闭双曲n-流形，证明了长度足够大的随机闭测地线具有指定宽度的领圈，且大部分此类测地线是简单的，正面回答了Kirby列表中的相关问题。
Let M be an orientable hyperbolic surface without boundary and let $γ$ be a closed geodesic in M. We prove that any side of any triangle formed by distinct lifts of $γ$ in H2 is shorter than $γ$.
We generalize the results of [AS], finding large classes of totally geodesic Seifert surfaces in hyperbolic knot and link complements, each the lift of a rigid 2-orbifold embedded in some hyperbolic 3-orbifold. In addition, we provide a uniqueness theorem and demonstrate that many knots cannot possess totally geodesic Seifert surfaces by giving bounds on the width invariant in the presence of such a surface. Finally, we utilize these examples to demonstrate that the Six Theorem is sharp for knot complements in the 3-sphere.
Let $Σ$ be a complete finite-area orientable hyperbolic surface with one cusp, and let $\mathcal{R}$ be the space of complete geodesic rays in $Σ$ emanating from the puncture. Then there is a natural action of the mapping class group of $Σ$ on $\mathcal{R}$. We show that this action is "almost everywhere" wandering.
We give sharp upper bounds on the injectivity radii of complete hyperbolic surfaces of finite area with some geodesic boundary components. The given bounds are over all such surfaces with any fixed topology; in particular, boundary lengths are not fixed. This extends the first author's result to the with-boundary setting. In the second part of the paper we comment on another direction for extending this result, via the systole of loops function.
Growth of Simple Closed Geodesics in Finite-Volume Hyperbolic Manifolds
针对闭双曲3-流形，研究其本原简单闭测地线数目随长度的增长规律，得出长度不超过L的该类测地线数目为e^((2+o(1))L)的结论
We prove that for any orientable connected surface of finite type which is not a a sphere with at most four punctures or a torus with at most two punctures, any homeomorphism of the space of geodesic laminations of this surface, equipped with the Thurston topology, is induced by a homeomorphism of the surface.
In this paper we show that totally geodesic subspaces determine the commensurability class of a standard arithmetic hyperbolic $n$-orbifold, $n\ge 4$. Many of the results are more general and apply to locally symmetric spaces associated to arithmetic lattices in $\mathbb{R}$-simple Lie groups of type $B_n$ and $D_n$. We use a combination of techniques from algebraic groups and quadratic forms to prove several results about these spaces.
Let M be a closed hyperbolic three manifold. We construct closed surfaces which map by immersions into M so that for each one the corresponding mapping on the universal covering spaces is an embedding, or, in other words, the corresponding induced mapping on fundamental groups is an injection.
We prove that if a closed hyperbolic 3-manifold M contains infinitely many totally geodesic surfaces, then M is arithmetic.
In this paper we consider finite volume hyperbolic manifolds X with non-empty totally geodesic boundary. We consider the distribution of the times for the geodesic flow to hit the boundary and derive a formula for the moments of the associated random variable in terms of the orthospectrum. We show that the the first two moments correspond to two cases of known identities for the orthospectrum. We further obtain an explicit formula in terms of the trilogarithm functions for the average time for the geodesic flow to hit the boundary in the surface case, using the third moment.
The space $\mathrm{GC} (Σ)$ of geodesic currents on a hyperbolic surface $Σ$ can be considered as a completion of the set of weighted closed geodesics on $Σ$ when $Σ$ is compact, since the set of rational geodesic currents on $Σ$, which correspond to weighted closed geodesics, is a dense subset of $\mathrm{GC}(Σ)$. We prove that even when $Σ$ is a cusped hyperbolic surface with finite area, $\mathrm{GC}(Σ)$ has the denseness property of rational geodesic currents, which correspond not only to weighted closed geodesics on $Σ$ but also to weighted geodesics connecting two cusps. In addition, we …
In this paper we derive explicit estimates for the functions which appear in the previous work of Bridgeman and Kahn. As a consequence, we obtain an explicit lower bound for the length of the shortest orthogeodesic in terms of the volume of a hyperbolic manifold with totally geodesic boundary. We also give an alternative derivation of a lower bound for the volumes of these manifolds as a function of the dimension.
We present a quantitative isolation property of the lifts of properly immersed geodesic planes in the frame bundle of a geometrically finite hyperbolic $3$-manifold. Our estimates are polynomials in the tight areas and Bowen-Margulis-Sullivan densities of geodesic planes, with degree given by the modified critical exponents.
Nonsimple closed geodesics with given intersection number on hyperbolic surfaces
该研究改进了双曲曲面上自相交数为k的闭测地线最小长度公式成立的阈值，将其从k>10^13350降至k>1750，且给出了独立于原有结论的证明。
We prove a quantitative estimate, with a power saving error term, for the number of simple closed geodesics of length at most $L$ on a compact surface equipped with a Riemannian metric of negative curvature. The proof relies on the exponential mixing rate for the Teichmüller geodesic flow.
We extend to the context of hyperbolic 3-manifolds with geodesic boundary Thurston's approach to hyperbolization by means of geometric triangulations. In particular, we introduce moduli for (partially) truncated hyperbolic tetrahedra, and we discuss consistency and completeness equations. Moreover, building on previous work of Ushijima, we extend Weeks' tilt formula algorithm, which computes the Epstein-Penner canonical triangulation, to an algorithm that computes the Kojima triangulation. The theory is particularly interesting in the case of complete finite-volume manifolds with geodesic boun…
Cutoff for geodesic paths on hyperbolic manifolds
本文证明任意紧双曲流形上从局部化初始条件出发的测地路径具有截止现象，并将Golubev和Kamber关于大体积双曲曲面的结果推广到任意维数，方法基于Lubetzky-Peres谱策略和球面平均算子的谱分析。
A well-known theorem of Wolpert shows that the Weil-Petersson symplectic form on Teichmüller space, computed on two infinitesimal twists along simple closed geodesics on a fixed hyperbolic surface, equals the sum of the cosines of the intersection angles. We define an infinitesimal deformation starting from a more general object, namely a balanced geodesic graph, by which any tangent vector to Teichmüller space can be represented. We then prove a generalization of Wolpert's formula for these deformations. In the case of simple closed curves, we recover the theorem of Wolpert.
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