Mobility spaces and geodesics for the n-sphere

Impossible configurations for geodesics on negatively-curved surfaces

Hass and Scott's example of a 4-valent graph on the 3-punctured sphere that cannot be realized by geodesics in any metric of negative curvature is generalized to impossible configurations filling surfaces of genus $n$ with $k$ punctures for any $n$ and $k$.

2-球面上具有给定区间内测地曲率的曲线空间的分支

On the components of spaces of curves on the 2-sphere with geodesic curvature in a prescribed interval

利用$κ_1$和$κ_2$，得到了2-球面上测地曲率约束在开区间$(κ_1,κ_2)$内的闭曲线空间的连通分支的简单刻画.建立了关于这些空间的拓扑的许多结果。特别是，我们确定的同胚类的一些组件。

泰希米勒空间中单位切球的凸结构

Convex structures of the unit tangent spheres in Teichm{ü}ller space

研究泰希米勒空间中单位切球的凸结构，通过分析瑟斯顿度量的芬斯勒无穷小球，得到面、暴露面和极点的特征，包括面与链回归测地线层叠的关系等，为相关拓扑部分提供了新方法并肯定了一个猜想。

Which geodesic flows are left-handed ?

We prove that the geodesic flow on the unit tangent bundle to a hyperbolic 2-orbifold is left-handed if and only if the orbifold is a sphere with three conic points. As a consequence, on the unit tangent bundle to a 3-conic sphere, the lift of every finite collection of closed geodesics that is zero in integral homology is the binding of an open book decomposition.

Gromoll-Meyer球面中的极小Brieskorn 5-球面及其应用

A minimal Brieskorn 5-sphere in the Gromoll-Meyer sphere and its applications

我们认为格罗莫尔-迈耶球面σ ^[7]是一条简单的闭测地线和一个极小子球面σ ^[5]的测地线连接，它可以等价地与布里斯科恩球面W ^[5_3]等同。作为应用，我们特别确定了Σ ^[7]的全等距群，对所有自由作用的闭子群进行了分类，确定了相应轨道空间的同伦类型，将5维的赫希-米尔诺尔对合与W ^[5_3]的卡拉比对合等同起来，并得到了Brieskorn球面W ^[5_3]和W ^[13_3]与标准欧几里得球面之间的同构的显式公式。

穿孔球面和超椭圆曲面的Bers常数

Bers' constants for punctured spheres and hyperelliptic surfaces

本文证明了Buser关于带尖点球面和超椭圆曲面的Bers常数的猜想。更具体地说，我们的主要定理指出，任何具有$n$尖点的双曲球面都有一个裤子分解，其所有长度的测地线都以$n$的一个常数的大致平方根为界。其他结果包括下和上界的Bers常数的超椭圆表面和领域的边界测地线。

随机双曲穿孔球面上的短测地线与小特征值

Short geodesics and small eigenvalues on random hyperbolic punctured spheres

本文研究了 $n\to\infty$ 时 Weil-Petersson 随机双曲穿孔球面上的短测地线与小特征值，证明闭测地线长度经缩放后收敛于 Poisson 点过程，并得出具有至少 $k=o(n)$ 个任意小特征值的概率趋于 $1$。

A characterization of Zoll Riemannian metrics on the 2-sphere

The simple length spectrum of a Riemannian manifold is the set of lengths of its simple closed geodesics. We prove a theorem claimed by Lusternik: in any Riemannian 2-sphere whose simple length spectrum consists of only one element L, any geodesic is simple closed with length L.

什么是几乎正态曲面

What is an Almost Normal Surface

拓扑算法理论的一个重大突破发生在1992年，当时Hyam Rubinstein提出了几乎法向曲面的概念。我们解释了如何几乎正常的表面自然出现的测地线和极小曲面的研究。稳定测地线和不稳定测地线的模式可以用来刻画曲面间的2-球面，类似的模式可以刻画3-流形间的3-球面。类似的模式正常和几乎正常的表面导致鲁宾斯坦的算法识别3球。

具有测地曲率的二维球面上曲线在给定区间内的同伦

Homotopies of Curves on the 2-Sphere with Geodesic Curvature in a Prescribed Interval

设$C_{k_1}^{k_2}$表示球面$S^2$上所有类$C^r$的闭曲线的集合，其测地曲率被限制在$(k_1,k_2)$中，具有$C^r$拓扑（对于某些$r >= 2$和可能无限的$k_1 < k_2$）。1970年，J.Little证明了具有正测地曲率的闭曲线空间$C_0^{+\infty}$有三个连通分支。设$r_i = arccot k_i$（i = 1，2）.我们发现，$C_{k_1}^{k_2}$有n个连接组件$C_1, ..., C_n$，其中n是最大的整数小于或等于$π/(r_1-r_2) + 1$，$C_j$包含循环遍历j次（$1 <= j <= n$）。对于任何自然数m，$C_{n-1}$分量也包含遍历$(n-1) + 2m$次的圆，$C_n$分量也包含遍历$n + 2m$次的圆。此外，每个$C_1, ..., C_{n-2}$同伦等价于$SO_3$（$n >= 3$）。一个简单的特征的组件的性质的曲线和证明$C_{k_1}^{k_2}$同胚$C_{k'_1}^{k'_2}$时，$r_1 - r_2 = r'_1 - r'_2$（$r'_i = arccot k'_i$）也提出。

Spherical complexities, with applications to closed geodesics

We construct and discuss new numerical homotopy invariants of topological spaces that are suitable for the study of functions on loop and sphere spaces. These invariants resemble the Lusternik-Schnirelmann category and provide lower bounds for the numbers of critical orbits of SO(n)-invariant functions on spaces of n-spheres in a manifold. Lower bounds on these invariants are derived using weights of cohomology classes. As an application, we prove new existence results for closed geodesics on Finsler manifolds of positive flag curvature satisfying a pinching condition.