On strictly convex subsets in negatively curved manifolds
Jouni Parkkonen, Frédéric Paulin
在完备单连通的负曲率紧致黎曼流形X中，利用C的边界的外曲率，给出了子集C是X的某个凸子集的ε-邻域的一个精确判据.
In a complete simply connected Riemannian manifold X of pinched negative curvature, we give a sharp criterion for a subset C to be the epsilon-neighbourhood of some convex subset of X, in terms of the extrinsic curvatures of the boundary of C.
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L2-cohomology of negatively curved Kaehler manifolds of finite volume
我们计算有限体积的负弯曲Kaehler流形上的$L^2$调和形式空间（中次以外）。
We classify closed, simply-connected, non-negatively curved 6-manifolds of almost maximal symmetry rank up to equivariant diffeomorphism.
Lp-cohomology of negatively curved manifolds
计算了某些负曲流形的$L^p$ -上同调空间。我们处理两种情况：流形有限体积和充分捏负曲率，共形紧流形。
Quasiconformality and geometrical finiteness in Carnot--Carathéodory and negatively curved spaces
本文概述了负曲空间中的等变拟共形和拟对称同胚以及非紧几何有限负曲流形及其无穷远处具有Carnot-Carathéodory结构的边界的几何和拓扑研究的最新进展，并阐述了几个公开问题。特别是，最有趣的是复杂的双曲流形与柯西-黎曼结构在无穷远，它占据了一个杰出的利基，其性质使他们令人惊讶地不同于真正的双曲。
This paper generalizes D. Burago and S. Ivanov's work on filling volume minimality and boundary rigidity of almost real hyperbolic metrics. We show that regions with metrics close to a negatively curved symmetric metric are strict minimal fillings and hence boundary rigid. This includes perturbations of complex, quaternionic and Cayley hyperbolic metrics.
We prove that the existence of one flat horosphere in the universal cover of a closed, strictly quarter pinched, negatively curved Riemannian manifold of dimension n with n greater than or equal to 3, implies that the manifold is homothetic to a real hyperbolic manifold.
Negatively curved Einstein metrics on Gromov-Thurston manifolds
本文为每个$n\geq 4$构造了无穷多个互不同伦的闭流形，它们具有负曲率爱因斯坦度量但无局部对称度量。
We construct radial fundamental solutions for the differential form Laplacian on negatively curved symmetric spaces. At least one of these Green's functions also yields a Biot-Savart Opearator, i.e. a right inverse of the exterior differential on closed forms with image in the kernel of the codifferential. Any Biot-Savart operator gives rise to a Gauss linking integral.
Heintze-Kobayashi-Wolf theory for negatively curved homogeneous Finsler manifolds
该研究将Heintze-Kobayashi-Wolf理论推广至齐性Finsler几何，证明连通负曲率齐性Finsler流形等距于单连通可解李群的左不变度量空间，且Heintze判别准则是实可解李代数生成对应李群的充要条件。
We prove that a quasi-isometric map, and more generally a coarse embedding, between pinched Hadamard manifolds is within bounded distance from a unique harmonic map.
We prove some rigidity results on geodesic orbit Finsler spaces with non-positive curvature. In particular, we show that a geodesic Finsler space with strictly negative flag curvature must be a non-compact Riemannian symmetric space of rank one.
Einstein metrics on some exotic negatively curved manifolds
这篇论文被撤回的作者由于一个错误的证明的主要结果;基本上参数R使用的证明可能取决于流形（M，g），而不仅仅是对维数和pinching常数。
Non-negatively curved 5-manifolds with almost maximal symmetry rank
我们证明了一个允许有效等距$T^2$作用的闭的、单连通的、非负曲的5-流形是$S^5$、$S^3\times S^2$、$S^3\tilde{\times} S^2$（$S^2$上的非平凡$S^3$ -丛）或Wu流形$SU(3)/SO(3)$中的一个流形.
We prove that if a closed, smooth, simply-connected 4-manifold with a circle action admits an almost non-negatively curved sequence of invariant Riemannian metrics, then it also admits a non-negatively curved Riemannian metric invariant with respect to the same action. The same is shown for torus actions of higher rank, giving a classification of closed, smooth, simply-connected 4-manifolds of almost non-negative curvature under the assumption of torus symmetry.
Mean curvature and compactification of surfaces in a negatively curved Cartan-Hadamard manifold
本文证明了一个Chern-Osserman型不等式，它涉及到完全曲面的体积增长，该曲面具有受控的中曲率，且适当地浸入Cartan-Hadamard流形$N$中，该流形的截面曲率从上到下由一个负量$K_{N}\leq b<0$限定
Motivated by a recent groundbreaking work of Ontaneda, we describe a sizable class of closed manifolds such that the product of each manifold in the class with the real line admits a complete metric of bounded negative sectional curvature which is an exponentially warped near one end and has finite volume near the other end.
The existence of negatively curved metrics on locally conformally flat manifolds with boundary
本文研究带边界流形上共形度量的存在性，借助 Morse 函数构造负截面曲率度量，并在无共形平坦性假设时构造正 Einstein 张量度量。
Let $(X, g_0)$ be a simply connected, complete, negatively curved Riemannian manifold. We prove local and infinitesimal rigidity results for compactly supported deformations of the metric $g_0$. For any negatively curved metric $g$ equal to $g_0$ outside a compact, the identity map of $X$ induces a natural boundary map between the boundaries at infinity of $X$ with respect to $g_0$ and $g$. We show that if $(g_t)$ is a smooth 1-parameter family of negatively curved metrics all equal to $g_0$ outside a fixed compact then if all the boundary maps (between the boundaries of $X$ with respect to $g…
The space of nonpositively curved metrics of a negatively curved manifold
我们证明了负弯曲流形的非正弯曲度量空间是高度非连通的。
Effective stability of negatively curved Einstein metrics in dimensions at least $4$
该研究证明了至少四维闭合流形在满足一定条件时存在真正的负截面曲率爱因斯坦度量，并且 pinching 常数不受直径、体积或注入半径的限制。
Cohomogeneity-two torus actions on non-negatively curved manifolds of low dimension
设$M^n$，$n \in \{4,5,6\}$是一个紧的单连通$n$ -流形，它允许一个具有非负曲率的黎曼度量和一个具有极大可能秩的等距群.然后任何光滑的，有效的作用在$M^n$上的环面$T^{n-2}$是等变同构的等距作用在正常的双商。此外，它遵循任何有效的，等距圆作用在一个紧凑的，单连通的，非负弯曲的四维流形是等变同构的一个有效的，等距作用在正常的双商。
We prove that a reversible Berwaldian Finsler structure $\left(M,F\right)$ with base-independent non-negative radial flag curvature and large volume growth does not have any closed geodesics.
Motivated by the study of Killing forms on compact Riemannian manifolds of negative sectional curvature, we introduce the notion of generalized vector cross products on $\mathbb{R}^n$ and give their classification. Using previous results about Killing tensors on negatively curved manifolds and a new characterization of $\mathrm{SU}(3)$-structures in dimension $6$ whose associated $3$-form is Killing, we then show that every Killing $3$-form on a compact $n$-dimensional Riemannian manifold with negative sectional curvature vanishes if $n\ge 4$.
Teichmüller Spaces and Bundles with Negatively Curved Fibers
我们研究了高维流形上的负弯曲度量的Teichmüller空间，并将其应用于具有负弯曲纤维的丛。
Let $(X, D)$ be a log smooth log canonical pair such that $K_X+D$ is ample. Extending a theorem of Guenancia and building on his techniques, we show that negatively curved Kähler-Einstein crossing edge metrics converge to Kähler-Einstein mixed cusp and edge metrics smoothly away from the divisor when some of the cone angles converge to $0$. We further show that near the divisor such normalized Kähler-Einstein crossing edge metrics converge to a mixed cylinder and edge metric in the pointed Gromov-Hausdorff sense when some of the cone angles converge to $0$ at (possibly) different speeds.
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.
Rigidity of non-negatively curved metrics on open five-dimensional manifolds
作为在关于$S^2 \times S^2$上不存在具有正截面曲率的度量的霍普夫猜想方向上的第一步，D.Gromoll和K.Tapp在[GT]中提出了以下建议（弱Hopf）猜想（关于$S^2 \times R^3$上的非负弯曲度量的刚性）：“具有任意完全度量的$S^2 \times B^3\subset S^2 \times R^3$的边界$S^2\times S^2$是非线性的，负截面曲率包含$S^2 \times S^2$的曲率消失的点”。在本说明中，我们证实了这一点。
Pinching estimates for negatively curved manifolds with nilpotent fundamental groups
设$M$是$[-a^2,-1]$中截面曲率的完备黎曼度量，其基本群包含有限指数的$k$ -步幂零子群.证明了$a\ge k$回答了M.格罗莫夫此外，我们表明，对于任何$ε>0$，流形$M$承认一个完整的黎曼度量的截面曲率内$[-(k+ε)^2,-1]$。
We prove the following entropy-rigidity result in finite volume: if $X$ is a negatively curved manifold with curvature $-b^2\leq K_X \leq -1$, then $Ent_{top}(X) = n-1$ if and only if $X$ is hyperbolic. In particular, if $X$ has the same length spectrum of a hyperbolic manifold $X_0$, the it is isometric to $X_0$ (we also give a direct, entropy-free proof of this fact). We compare with the classical theorems holding in the compact case, pointing out the main difficulties to extend them to finite volume manifolds.
Fenchel-Willmore and Sobolev-type inequalities for submanifolds in non-negatively curved manifolds
本文建立Fenchel-Willmore不等式与对数Sobolev不等式的联系，推广至带边界及完备非紧子流形，并导出尖锐Sobolev型不等式，应用于曲面拓扑。
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。
In this paper, we show that any compact K$ä$hler manifold homotopic to a compact Riemannian manifold with negative sectional curvature admits a K$ä$hler-Einstein metric of general type. Moreover, we prove that, on a compact symplectic manifold $X$ homotopic to a compact Riemannian manifold with negative sectional curvature, for any almost complex structure $J$ compatible with the symplectic form, there is no non-constant $J$-holomorphic entire curve $f:C \rightarrow X$.
Negatively curved Kähler metrics on total spaces of a class of vector bundles
本文针对一类向量丛全空间，在其配备非正曲率Kähler度量的条件下，证明其上存在丰富的负全纯双截曲率完备Kähler度量，并给出全纯函数维数估计与全纯映射Liouville定理。
Let $X, Y$ be complete, simply connected Riemannian surfaces with pinched negative curvature $-b^2 \leq K \leq -1$. We show that if $f : \partial X \to \partial Y$ is a Moebius homeomorphism between the boundaries at infinity of $X, Y$, then $f$ extends to an isometry $F : X \to Y$. This can be viewed as a generalization of Otal's marked length spectrum rigidity theorem for closed, negatively curved surfaces, in the sense that Otal's theorem asserts that if $X, Y$ admit properly discontinuous, cocompact, free actions by groups of isometries and the boundary map $f$ is Moebius and equivariant w…
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 note, we study the radius of positively curved or non-negatively curved Alexandrov space with strictly convex boundary, with convexity measured by the Base-Angle defined by Alexander and Bishop. We also estimate the volume of the boundary of non-negatively curved spaces as well as the rigidity case, which can be thought as a non-negatively curved version of a recent result of Grove-Petersen.
Non-negatively Curved Manifolds with Maximal Symmetry Rank in Low Dimensions
我们分类封闭，单连通非负弯曲5-流形承认（几乎）有效，等距$T^3$或$T^2$行动。作为一个直接的后果，我们表明，对于任何流形，尺寸高达9，并包括在相同的假设下，最大对称秩等于$[2n/3]$和自由秩是小于或等于一半的值。
Complete negatively curved immersed ends in $\Bbb R^3$
本文将著名的Efimov定理推广到$\real^3$中的浸入端点.更精确地说，设$M$是一个具有紧边界的非紧连通曲面。则不存在满足$\int_M |K|=+\infty$和$K\le-κ<0$的$M$到$\Bbb R^3$的完全等距浸入，其中$κ$是正常数，$K$是$M$的高斯曲率。特别是Efimov定理适用于完整的Hadamard浸入曲面，其高斯曲率$K$在紧集之外是有界的。
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.
Non-negatively curved metrics on $S^2\times S^2$ admitting Killing fields
%auto-ignore这篇论文由于一个重大错误而被作者撤回。
In this note it is shown that Berwald spaces admitting the same norm-preserving torsion-free affine connection have the same (weighted) Ricci curvatures. Combing this with Szabó's Berwald metrization theorem one can apply the Cheeger-Gromoll splitting theorem in order to get a full structure theorem for Berwald spaces of non-negative Ricci curvature. Furthermore, if none of the factor is a symmetric space one obtains an explicit expression of Finsler norm of the resulting product. By the general structure theorem one can apply the soul theorem to the factor in case of non-negative flag curvatu…
In this short note we survey some results about the fundamental group of a compact negatively curved manifold. In particular, we review a theorem of Gusevskij, it states that the fundamental group of a compact negatively curved manifold does not belong to $\mathcal{C},$ where $\mathcal{C}$ is the smallest class of groups that contains all amenable groups and is closed under free products and finite extensions.
On the spectrum of a finite-volume negatively-curved manifold
我们证明了具有有界截面曲率的非紧流形，其端点足够接近于射线的Gromov-Hausdorff，具有有限维的平方可积调和形式空间。在有限体积流形的特殊情况下，我们证明了p-型Laplacian的本质谱是与端点相关的普通微分算子集合的本质谱的并集.我们给出了这样的流形的例子与曲率捏任意接近-1，并与无限数量的差距的频谱中的功能拉普拉斯。
We constructed physically stable sp2 negatively curved cubic carbon structures which reticulate a Schwarz P-like surface. The method for constructing such crystal structures is based on the notion of the standard realization of abstract crystal lattices. In this paper, we expound on the mathematical method to construct such crystal structures.
Topology of non-negatively curved manifolds
我们研究了几类流形，它们具有相同的上同调环作为一个空间（一个家庭的双折射，这是一个主要的来源流形的正曲率）。一类是CP ^2上的3-球面丛。另一个是基上的圆丛，它本身是CP^2上的CP^1丛族之一。我们分类这样的流形的同胚使用Kreck Stolz不变量。比较的不变量，然后用来找到许多的全空间，这些丛的正弯曲的Eschenburg空间的同构。每个丛的全空间（在丛不自旋的情况下）也允许有非负截面曲率的度量，在某些情况下也允许有爱因斯坦度量。
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.
Cheeger constant rigidity for cocompact negatively curved manifolds
本文证明在闭连通负曲率流形的万有覆盖上，Cheeger常数达到下界$m-1$时，流形必等距于双曲空间$\Hh^m(-1)$，从而给出刚性结论。
Answering a question by Margulis we prove that the conclusion of Selberg's Lemma fails for discrete isometry groups of negatively curved Hadamard manifolds.
On finite volume, negatively curved manifolds
我们研究非紧的、完备的、有限体积的、负弯曲的流形$M$。我们构造$M$与无限生成的基本群在所有维度$n \geq 2$。我们构造了$M$，它的尖点截面是全维$n\geq 3$中的紧双曲流形.相反，如果截面曲率为$-1<K(M)<0$，则尖点截面的单纯体积为零。我们构造了不包含任何抛物等距的负弯曲格。我们证明了存在$M$使得$\widetilde{M}$不满足可见性公理。我们给出了一个条件的曲率增长与体积衰减，保证拓扑有限性。本文对有限体积负曲流形提出几个问题。
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。
On Negatively Curved Finsler Manifolds of Scalar Curvature
本文证明了维数n>2的紧致流形上负曲Finsler度量的一个整体刚性定理。证明了对于这样的Finsler流形，如果旗曲率是切丛上的标量函数，则Finsler度量是Randers型的.我们还研究了Finsler度量是局部射影平坦的情况。
Let $M$ be a graph manifold such that each piece of its JSJ decomposition has the $\Bbb H^2 \times \Bbb R$ geometry. Assume that the pieces are glued by isometries. Then, there exists a complete Riemannian metric on $\Bbb R \times M$ which is an "eventually warped cusp metric" with the sectional curvature $K$ satisfying $-1 \le K <0$. A theorem by Ontaneda then implies that $M$ appears as an end of a 4-dimensional, complete, non-compact Riemannian manifold of finite volume with sectional curvature $K$ satisfying $-1 \le K <0$.
Hyperplane arrangements in negatively curved manifolds and relative hyperbolicity
我们证明了由负曲率流形中的超平面安排所产生的某些非球面流形具有相对双曲的基本群。
Let $n\ge 2$ and $k\ge 1$ be two integers. Let $M$ be an isometrically immersed closed $n$-submanifold of co-dimension $k$ that is homotopic to a point in a complete manifold $N$, where the sectional curvature of $N$ is no more than $δ<0$. We prove that the total squared mean curvature of $M$ in $N$ and the first non-zero eigenvalue $λ_1(M)$ of $M$ satisfies $$λ_1(M)\le n\left(δ+\frac{1}{\operatorname{Vol} M}\int_M |H|^2 \operatorname{dvol}\right).$$ The equality implies that $M$ is minimally immersed in a metric sphere after lifted to the universal cover of $N$. This completely settles an ope…
We survey several notions of entropy related to a compact manifold of negative curvature, some relations between them, and the rigidity problems.
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 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.
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。
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