Jayadev S. Athreya, Steven P. Lalley, Jenya Sapir, Matthew Wroten
It is shown that the tessellation of a compact, negatively curved surface induced by a typical long geodesic segment, when properly scaled, looks locally like a Poisson line process. This implies that the global statistics of the tessellation -- for instance, the fraction of triangles -- approach those of the limiting Poisson line process.
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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 show that the space of nonpositively curved metrics of a negatively curved manifold is highly non connected.
We study the Teichmüller space of negatively curved metrics on a high dimensional manifold, with applications to bundles with negatively curved fibers.
The existence of essential closed surfaces surfaces is proven for finite coverings of 3-manifolds that are triangulated by finitely many topological ideal tetrahedra and admit a regular, negatively curved, ideal structure.
On the topology of the space of negatively curved metrics
证明了闭负曲黎曼$n$流形$n\geq 10$的负曲度量空间是高度非连通的.
On the moduli space of negatively curved metrics of a hyperbolic manifold
研究了双曲流形的负曲度量的模空间。
A caveat on the convergence of the Ricci flow for pinched negatively curved manifolds
我们给出的例子，捏负曲流形的里奇流不收敛顺利。
Conformal currents and the entropy of negatively curved three-manifolds
该研究通过刻画闭双曲三维流形上测地流与共形流的交集，得到负曲率度量相关熵与面积比的精确界，并给出三维莫斯托刚性定理的新证明。
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.
Negatively Ricci curved manifolds
在本文中，我们宣布了以下结果：``每个维数为$\ge3$的流形允许一个完全的负Ricci曲度量。此外，我们描述了一些更清晰的结果和草图证明。
In this paper we prove that for all $n=4k-2$, $k\ge2$ there exists closed $n$-dimensional Riemannian manifolds $M$ with negative sectional curvature that do not have the homotopy type of a locally symmetric space, such that $π_{1}(\mathcal{T}^{<0}(M))$ is non-trivial. $\mathcal{T}^{<0}(M)$ denotes the Teichmüller space of all negatively curved Riemannian metrics on $M$, which is the topological quotient of the space of all negatively curved metrics modulo the space of self-diffeomorphisms of $M$ that are homotopic to the identity. Gromov Thurston branched cover manifolds provide examples of ne…
A simplicial complex is called negatively curved if all its simplices are isometric to simplices in hyperbolic space, and it satisfies Gromov's Link Condition. We prove that, subject to certain conditions, a compact graph of spaces whose vertex spaces are negatively curved 2-complexes, and whose edge spaces are points or circles, is negatively curved. As a consequence, we deduce that certain groups are CAT(-1). These include hyperbolic limit groups, and hyperbolic groups whose JSJ components are fundamental groups of negatively curved 2-complexes---for example, finite graphs of free groups wit…
We prove by an algebraic method that the embedding of the Teichmuller space in the space of geodesic currents is totally linearly independent. We prove a similar result for all negatively curved surfaces using an ergodic argument.
We study the bounded fundamental class in the top dimensional bounded cohomology of negatively curved manifolds with infinite volume. We prove that the bounded fundamental class of $M$ vanishes if $M$ is geometrically finite. Furthermore, when $M$ is a $\mathbb{R}$-rank one locally symmetric space, we show that the bounded fundamental class of $M$ vanishes if and only if the Riemannian volume form on $M$ is the differential of a bounded differential form on $M$.
Answering a question by Margulis we prove that the conclusion of Selberg's Lemma fails for discrete isometry groups of negatively curved Hadamard manifolds.
In this paper we prove that for all $n=4k-2$, $k\ge2$ there exists a closed smooth complex hyperbolic manifold $M$ with real dimension $n$ having non-trivial $π_1(\mathcal{T}^{<0}(M))$. $\mathcal{T}^{<0}(M)$ denotes the Teichmüller space of all negatively curved Riemannian metrics on $M$, which is the topological quotient of the space of all negatively curved metrics modulo the space of self-diffeomorphisms of $M$ that are homotopic to the identity.
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 $k$.
Let $Υ$ be a compact, negatively curved surface. From the (finite) set of all closed geodesics on $Υ$ of length $\leq L$, choose one, say $γ_{L}$, at random and let $N (γ_{L})$ be the number of its self-intersections. It is known that there is a positive constant $κ$ depending on the metric such that $N (γ_{L})/L^{2} \rightarrow κ$ in probability as $L\rightarrow \infty$. The main results of this paper concern the size of typical fluctuations of $N (γ_{L})$ about $κL^{2}$. It is proved that if the metric has constant curvature -1 then typical fluctuations are of order $L$, in particular, $(N (…
Let $g_t$ be a smooth 1-parameter family of negatively curved metrics on a closed hyperbolic 3-manifold $M$ starting at the hyperbolic metric. We construct foliations of the Grassmann bundle $Gr_2(M)$ of tangent 2-planes whose leaves are (lifts of) minimal surfaces in $(M,g_t)$. These foliations are deformations of the foliation of $Gr_2(M)$ by (lifts of) totally geodesic planes projected down from the universal cover $\mathbb{H}^3$. Our construction continues to work as long as the sum of the squares of the principal curvatures of the (projections to $M$) of the leaves remains pointwise small…
We prove that the space of complete, finite volume, pinched negatively curved Riemannian metrics on a smooth high-dimensional manifold is either empty or it is highly non-connected, provided their behavior at infinity is similar.
Veering branched surfaces, surgeries, and geodesic flows
本文引入转向分支曲面作为研究转向三角剖分的对偶方法，通过手术操作显式构造了对应于负曲率曲面测地流的转向分支曲面，并给出了测地流的显式Markov分割及钻孔单位切丛的分类。
We prove that the Teichmüller space $\mathcal{T}^{<0}(M)$ of negatively curved metrics on a hyperbolic manifold $M$ has nontrivial $i$-th rational homotopy groups for some $i> \dim M$. Moreover, some elements of infinite order in $π_i B\mbox{Diff}(M)$ can be represented by bundles over $S^i$ with fiberwise negatively curved metrics.
Quasiisometries between negatively curved Hadamard manifolds
设X，Y是两个具有负截面曲率的紧致黎曼流形（维数不等于4）的泛覆盖.然后他们之间的每一个拟等距谎言在一个有限的距离bilipschitz同胚。
Sparse spectrally rigid sets for negatively curved manifolds
本文证明了在具有严格负截面曲率的紧致黎曼流形上，存在任意稀疏的谱刚性集合，且这些集合在基本群的每个方向上均有分布，从而揭示了标记长度谱在稀疏子集上的刚性特征。
We study noncompact, complete, finite volume, negatively curved manifolds $M$. We construct $M$ with infinitely generated fundamental groups in all dimensions $n \geq 2$. We construct $M$ whose cusp cross sections are compact hyperbolic manifolds in all dimension $n\geq 3$. In contrast we show that if sectional curvature $-1<K(M)<0$, then cusp cross sections have zero simplicial volume. We construct negatively curved lattices that do not contain any parabolic isometries. We show that there are $M$ such that $\widetilde{M}$ does not satisfy the visibility axiom. We give a condition on the curva…
In this paper we study non-negatively curved and rationally elliptic GKM$_4$ manifolds and orbifolds. We show that their rational cohomology rings are isomorphic to the rational cohomology of certain model orbifolds. These models are quotients of isometric actions of finite groups on non-negatively curved torus orbifolds. Moreover, we give a simplified proof of a characterisation of products of simplices among orbit spaces of locally standard torus manifolds. This characterisation was originally proved in [Wiemeler, Torus manifolds and non-negative curvature, arXiv:1401.0403] and was used ther…
A note on the characteristic classes of negatively curved manifolds
给出了关于紧致非正曲局部对称空间的一些著名结果的概念性证明。我们讨论了这些局部对称空间的Pontrjagin数的零与非零以及欧拉示性数。我们还建立消失的结果（有限覆盖）的Gromov-Thurston负弯曲流形的例子的Stiefel-Whitney数。我们提到了一些几何推论：Minnesota问题，一个下界覆盖度的切线映射到非负弯曲的顶点，并估计某些表示的复杂性，某些均匀格。
Approximation of recurrence in negatively curved metric spaces
对曲率小于或等于x，x<0的度量空间，证明了递归测地线可以用闭测地线逼近。提供了一个反例。
The Teichmüller Space of Pinched Negatively Curved Metrics on a Hyperbolic Manifold is not Contractible
对于光滑流形$M$，我们定义了$M$上所有黎曼度量的Teichmüller空间$\cT(M)$和$M$上$ε$ -pinched负曲度量的Teichmüller空间$\cT^ε(M)$，其中$0\leqε\leq\infty$ .我们证明，如果$M$是双曲的自然包含$\cT^ε(M)\hookrightarrow\cT(M)$，在一般情况下，不是同伦平凡的。特别地，$\cT^ε(M)$通常是不可收缩的。
We use classical results in smoothing theory to extract information about the rational homotopy groups of the space of negatively curved metrics on a high dimensional manifold. It is also shown that smooth M-bundles over spheres equipped with fiberwise negatively curved metrics, represent elements of finite order in the homotopy groups of the classifying space for smooth M-bundles, provided the dimension of M is large enough.
Dehn surgery and negatively curved 3-manifolds
证明了对任意给定的3-流形M，在3-球面和分数p/q（q > 22）中至多存在1/2个双曲纽结K，使得M是沿K通过p/q手术得到的.这是下列结果的推论。如果M是通过Dehn填充双曲三维流形X的尖点而得到的，其中每个填充斜率的长度大于2 π+ ε，则对于任何给定的M和ε> 0，X和填充斜率只有100多种可能性。在本文中，我们还研究了边界斜率的长度，以及给定的3-流形上的负曲度量序列。
Smooth Volume Rigidity for Manifolds with Negatively Curved Targets
建立了光滑闭流形与维数大于大于4的负曲流形之间的非零度连续映射同伦于光滑覆盖的条件，特别是当度为1时，该映射是同胚映射.当两个流形的体积或熵体积相差小于一个均匀常数后，适当的归一化的度量的条件成立。在依赖性是必要的这个意义上，结果是定性的。我们给出一些推论。
Counting open negatively curved manifolds up to tangential homotopy equivalence
在群G的适当假设下，证明了具有一致有界负截面曲率且基本群同构于群G的完备黎曼n-流形类分裂成100多个切同伦类型.因此，许多非球面流形不允许具有规定曲率界的完全负曲度量。
The classic 2pi-Theorem of Gromov and Thurston constructs a negatively curved metric on certain 3-manifolds obtained by Dehn filling. By Geometrization, any such manifold admits a hyperbolic metric. We outline a program using cross curvature flow to construct a smooth one-parameter family of metrics between the "2pi-metric" and the hyperbolic metric. We make partial progress in the program, proving long-time existence, preservation of negative sectional curvature, curvature bounds, and integral convergence to hyperbolic for the metrics under consideration.
Let $N$ be a smooth manifold that is homeomorphic but not diffeomorphic to a closed hyperbolic manifold $M$. In this paper, we study the extent to which $N$ admits as much symmetry as $M$. Our main results are examples of $N$ that exhibit two extremes of behavior. On the one hand, we find $N$ with maximal symmetry, i.e. Isom($M$) acts on $N$ by isometries with respect to some negatively curved metric on $N$. For these examples, Isom($M$) can be made arbitrarily large. On the other hand, we find $N$ with little symmetry, i.e. no subgroup of Isom($M$) of "small" index acts by diffeomorphisms of …
Let $M$ be a negatively curved compact Riemannian manifold with (possibly empty) convex boundary. Every closed differential $2$-form $ξ\inΩ^2(M)$ defines a bounded cocycle $c_ξ\in C_b^2(M)$ by integrating $ξ$ over straightened $2$-simplices. In particular Barge and Ghys proved that, when $M$ is a closed hyperbolic surface, $Ω^2(M)$ injects this way in $H_b^2(M)$ as an infinite dimensional subspace. We show that any class of the form $[c_ξ]$, where $ξ$ is an exact differential 2-form, belongs to the radical of the cup product on the graded algebra $H_b^\bullet(M)$.
Betti Numbers of Negatively Curved Orbifolds with Coefficients in Arbitrary Fields
该研究证明了负曲率轨道空间的贝蒂数随体积线性增长，并扩展了格罗莫夫的定理至轨道空间，解决了萨梅特关于任意特征的问题。
Quasiconformal Rigidity of Negatively Curved Three Manifolds
本文研究了具有截面曲率$-b^2\le K\le -1$的无限体积三维流形的刚性，并讨论了它的生成基本群。特别地，我们将具有空耗散集的非线性生成基本群的Sullivan拟共形刚性推广到负变曲率三维流形。我们还推广了Hamenstädt或最近Besson-Courtois-Gallot的刚性，3-流形具有无限体积和几何无限的基本群。
Finite automorphisms of negatively curved Poincare Duality groups
本文证明了：如果G是有限p-群（p素），G通过自同构作用在$δ$ -双曲Poincare对偶群上，则G的固定子群是Z/p上的Poincare对偶群，并举例说明G的固定子群甚至可能不是Z上的对偶群.
This is a survey on known results and open problems about Smooth and PL-Rigidity Problem for negatively curved locally symmetric spaces. We also review some developments about studying the basic topological properties of the space of negatively curved Riemannian metrics and the Teichmuller space of negatively curved metrics on a manifold.
We study noncompact, complete, finite volume, Riemannian 4-manifolds $M$ with sectional curvature $-1<K<0$. We prove that $π_1 M$ cannot be a 3-manifold group. A classical theorem of Gromov says that $M$ is homeomorphic to the interior of a compact manifold $\M$ with boundary $\partial\barM$. We show that for each $π_1$-injective boundary component $C$ of $\M$, the map $i_*$ induced by inclusion $i\colon C\rightarrow \M$ has infinite index image $i_*(π_1 C)$ in $π_1 \M$. We also prove that $M$ cannot be homotoped to be contained in $\partial\M$.
Mapping class groups of negatively curved four-manifolds
本文证明所有具严格负截面曲率度量的闭定向光滑四流形的 Torelli 群（光滑与拓扑版本）的第一有理同调群均为无限维，推广了 Budney--Gabai 的实双曲流形结果。
The bounded area class of negatively curved surfaces
该研究针对带挤压负曲率的定向曲面，证明其面积形式在二阶有界上同调中（除圆盘、圆柱外）非平凡，还揭示有界面积类对紧致曲面可识别常曲率度量、对无限型曲面无法区分非等距双曲结构的特性。
Tubes and Steklov eigenvalues in negatively curved manifolds
本文研究带全测地边界的紧致夹负曲率流形上的 Steklov 特征值，通过边界邻域 Steklov-Dirichlet 估计和管状邻域定理，建立第一非零特征值的体积型几何下界。
We study spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds. We construct, in particular, the first classes of manifolds for which these moduli spaces have non-trivial rational homotopy, homology and cohomology groups. We also show that in every dimension at least seven (respectively, at least eight) there exist closed (respectively, open) manifolds for which the space and moduli space of Riemannian metrics with non-negative sectional curvature has infinitely many path components. A completely analogous statem…
Bounded volume class and Cheeger isoperimetric constant for negatively curved manifolds
研究负曲率流形中有界体积类与Cheeger等周常数的关系，证明有界体积类消失当且仅当Cheeger常数为正，解决Kim和Kim的猜想部分。
Branched covers of hyperbolic manifolds and harmonic maps
我们给出了具有特殊性质的负曲流形之间的调和映射的例子。这些负弯曲流形不具有局部对称空间的同伦型。
Topology of closed asymptotic curves on negatively curved surfaces
受Nirenberg问题启发，研究负曲率曲面上闭渐近曲线的拓扑，利用Călugăreanu定理给出环绕数公式，并推广了相关观察，构造了法向单射但环绕数非零的例子。
Involutions of negatively curved groups with wild boundary behavior
我们考虑对（X，Y），其中X是一个紧的，局部CAT（-1）空间，Y是一个全测地子空间。包含诱导嵌入的边界在无穷大的通用覆盖，我们专注于这些领域的情况下，其尺寸相差2。我们表明，如果嵌入是驯服，那么它是unknotted。我们给出的例子对嵌入是打结（并可以实现为不动点集的对合）。我们还给出了高维（>5）球面中余维2球纽结性的一个判据。对δ-双曲群的相应结果也进行了讨论。即使在黎曼流形的情况下，一些推论也给出了新的结果。
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