Paul Norbury, J. Hyam Rubinstein
We consider the problem of finding embedded closed geodesics on the two-sphere with an incomplete metric defined outside a point. Various techniques including curve shortening methods are used.
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We show that cusped finite-volume hyperbolic 3-manifolds contain infinitely many simple closed geodesics.
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 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.
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。
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。
Using geodesic length functions, we define a natural family of real codimension 1 subvarieties of Teichmüller space, namely the subsets where the lengths of two distinct simple closed geodesics are of equal length. We investigate the point set topology of the union of all such hypersurfaces using elementary methods. Finally, this analysis is applied to investigate the nature of the Markoff conjecture.
In this paper, we use Chas-Sullivan theory on loop homology and Leray-Serre spectral sequence to investigate the topological structure of the non-contractible component of the free loop space on the real projective spaces with odd dimensions. Then we apply the result to get the resonance identity of non-contractible homologically visible prime closed geodesics on such spaces provided the total number of distinct prime closed geodesics is finite.
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。
We give a lower bound on the number of non-simple closed curves on a hyperbolic surface, given upper bounds on both length and self-intersection number. In particular, we carefully show how to construct closed geodesics on pairs of pants, and give a lower bound on the number of curves in this case. The lower bound for arbitrary surfaces follows from the lower bound on pairs of pants. This lower bound demonstrates that as the self-intersection number $K = K(L)$ goes from a constant to a quadratic function of $L$, the number of closed geodesics transitions from polynomial to exponential in $L$. …
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 give bounds on the number of non-simple closed curves on a negatively curved surface, given upper bounds on both length and self-intersection number. In particular, it was previously known that the number of all closed curves of length at most $L$ grows exponentially in $L$. We get exponentially tighter bounds given weak conditions on self-intersection number.
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.
Geometry and topology of closed geodesics complements in the 3-torus
本文研究了3-环面中闭测地线补空间的几何与拓扑，证明了至多三条方向线性无关闭测地线补空间的同胚类型由其方向子空间在 $PSL_3(\mathbb{Z})$ 作用下的轨道决定，并给出了依赖于Farey图距离的渐近锐利体积界限。
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 $2\arccosh(2k+1)$.
The goal of the chapter is to present certain aspects of the relationship between the study of simple closed geodesics and Teichmüller spaces.
On a hyperbolic Riemann surface, given two simple closed geodesics that intersect $n$ times, we address the question of a sharp lower bound $L_n$ on the length attained by the longest of the two geodesics. We show the existence of a surface $S_n$ on which there exists two simple closed geodesics of length $L_n$ intersecting $n$ times and explicitly find $L_n$ for $n\leq 3$.
We study embedding of closed totally geodesic hyperbolic 2-orbifolds in the Bianchi orbifolds $\mathbb{H}^3/PSL(2,\mathcal{O}_d)$. Our main result shows that there is a constant $c$ such that for $d$ large enough there are at least $cd$ closed embedded totally geodesic hyperbolic 2-orbifolds. Moreover we provide a list which conjecturally consists of those $d$ for which $\mathbb{H}^3/PSL(2,\mathcal{O}_d)$ does not contain a closed embedded totally geodesic hyperbolic 2-orbifold.
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.
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 $k$.
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.
In this note, we prove the existence of a closed geodesic of positive length on any compact developable orbifold of dimension 3, 5, or 7. The argument uses the stratification of the singular locus, and reduces the problem of existence of a closed geodesic on a compact developable orbifold to the case of even dimensional orbifolds with zero-dimensional singular locus and orbifold fundamental group infinite torsion and of odd exponent.
Let $X$ be a compact, geodesically complete, locally CAT(0) space such that the universal cover admits a rank one axis. Assume $X$ is not homothetic to a metric graph with integer edge lengths. Let $P_t$ be the number of parallel classes of oriented closed geodesics of length $\le t$; then $\lim\limits_{t \to \infty} P_t / \frac{e^{ht}}{ht} = 1$, where $h$ is the entropy of the geodesic flow on the space $SX$ of parametrized unit-speed geodesics in $X$.
Closed geodesics in dilation surfaces
本研究证明了任意膨胀曲面上闭测地线的方向在圆周上稠密，通过研究模空间中泰希米勒流作用下膨胀曲面Delaunay三角剖分的退化完成证明，推进了膨胀曲面几何的相关研究。
We compute the asymptotics, as R tends to infinity, of the number of closed geodesics in Moduli space of length at most R, or equivalently the number of pseudo-Anosov elements of the mapping class group of translation length at most R.
Let S be an oriented surface of genus g with m punctures. If 3g-3+m is at least 4 then we construct for every compact subset K of moduli space a closed Teichmueller geodesic not intersecting K.
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。
We show that the geodesic period spectrum of a Riemannian 2-orbifold all of whose geodesics are closed depends, up to a constant, only on its orbifold topology and compute it. In the manifold case we recover the fact proved by Gromoll, Grove and Pries that all prime geodesics have the same length. In the appendix we partly strengthen our result in terms of conjugacy of contact forms and explain how to deduce rigidity on the real projective plane based on a systolic inequality due to Pu. (We do not use a Lusternik-Schnirelmann type theorem on the existence of at least three simple closed geodes…
We give in this paper bounds for the Morse indices of a large class of simple geodesics on a surface with a generic metric. To our knowledge these bounds are the first that use only the generic hypothesis on the metric.
Our main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer $k$, we are interested in the set of all closed geodesics with at least $k$ (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We prove that their intersection numbers are upper bounded by a universal linear function in $k$ (which holds for any hyperbolic surface). Moreover, in the presence of cusps, we get bounds which imply that the self-intersection numbers behave asymptotically like $k$ for growing $k$.
Cutting a hyperbolic surface X along a simple closed multi-geodesic results in a hyperbolic structure on the complementary subsurface. We study the distribution of the shapes of these subsurfaces in moduli space as boundary lengths go to infinity, showing that they equidistribute to the Kontsevich measure on a corresponding moduli space of metric ribbon graphs. In particular, random subsurfaces look like random ribbon graphs, a law which does not depend on the initial choice of X. This result strengthens Mirzakhani's famous simple closed multi-geodesic counting theorems for hyperbolic surfaces…
We investigate the geometry of word metrics on fundamental groups of manifolds associated with the generating sets consisting of elements represented by closed geodesics. We ask whether the diameter of such a metric is finite or infinite. The first answer we interpret as an abundance of closed geodesics, while the second one as their scarcity. We discuss examples for both cases.
We obtain bounds on the numbers of intersections between triangulations as the conformal structure of a surface varies along a Teichm{ü}ller geodesic contained in an $\mathrm{SL}\left(2,\mathbb{R}\right)$-orbit closure of rank 1 in the moduli space of Abelian differentials. For $0 \leq θ\leq 1$, we obtain an exponential bound on the number of closed geodesics in the orbit closure, of length at most $R$, that spend at least $θ$-fraction of their length in a region with short saddle connections.
We use the $η$-invariant of Atiyah-Patodi-Singer to compute the Eells-Kuiper invariant for the Eells-Kuiper quaternionic projective plane. By combining with a known result of Bérard-Bergery, it shows that every Eells-Kuiper quaternionic projective plane carries a Riemannian metric such that all geodesics passing through a certain point are simply closed and of the same length.
Intersection Points of Closed Geodesics on Hyperbolic Surfaces of Finite Area
本文研究有限面积完备双曲曲面上闭测地线交点的空间分布，证明随长度阈值 T 趋于无穷，这些交点在曲面上趋于等分布。
In this paper, we show a local energy convexity of $W^{1,2}$ maps into $CAT(K)$ spaces. This energy convexity allows us to extend Colding and Minicozzi's width-sweepout construction to produce closed geodesics in any closed Alexandrov space of curvature bounded from above, which also provides a generalized version of the Birkhoff-Lyusternik theorem on the existence of non-trivial closed geodesics in the Alexandrov setting.
Length of a closed geodesic in 3-manifolds of positive scalar curvature
本文证明任何数量曲率不小于6的闭三维黎曼流形都含有一条长度小于22500的非平凡闭测地线，从而证实了Gromov在三维的猜想。
We prove a central limit theorem for the length of closed geodesics in any compact orientable hyperbolic surface. In the special case of a hyperbolic pair of pants, this settles a conjecture of Chas-Li-Maskit.
We apply topological methods and a Lusternik-Schnirelmann-type approach to prove existence results for closed geodesics of Finsler metrics on spheres and projective spaces. The main tool in the proofs are spherical complexities, which have been introduced in earlier work of the author. Using them, we show how pinching conditions and inequalities between a Finsler metric and a globally symmetric metric yield the existence of multiple closed geodesics as well as upper bounds on their lengths.
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。
Equidistribution of partial coverings from closed geodesics
研究有限体积双曲曲面中，由定向闭测地线集合关联的部分覆盖的等分布问题，通过证明给定在单位切丛中等分布的闭测地线集合，其相关部分覆盖也等分布，在模群情况给出等分布结果的另证。
The goal of this survey is to give a list of resent results about topology of manifolds admitting different metrics with the same geodesics. We emphasize the role of the theory of integrable systems in obtaining these results.
We show that on any translation surface, if a regular point is contained in a simple closed geodesic, then it is contained in infinitely many simple closed geodesics, whose directions are dense in the unit circle. Moreover, the set of points that are not contained in any simple closed geodesic is finite. We also construct explicit examples showing that such points exist. For a surface in any hyperelliptic component, we show that this finite exceptional set is actually empty. The proofs of our results use Apisa's classifications of periodic points and of $\GL(2,\R)$ orbit closures in hyperellip…
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