Igor Belegradek, Vitali Kapovitch
We give a diffeomorphism classification of pinched negatively curved manifolds with amenable fundamental groups, namely, they are precisely the Möbius band, and the products of a line with the total spaces of flat vector bundles over closed infranilmanifolds.
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Three-manifolds with non-negatively pinched Ricci curvature
研究非负曲率pinching的三维流形的Ricci流解，证明其具有标度不变曲率衰减和pinching保持，结合Lott和Deruelle-Schulze-Simon的工作，证明Hamilton的pinching猜想。
We give examples of pinched negatively curved manifolds for which the Ricci flow does not converge smoothly.
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.
For a smooth manifold $M$ we define the Teichmüller space $\cT(M)$ of all Riemannian metrics on $M$ and the Teichmüller space $\cT^ε(M)$ of $ε$-pinched negatively curved metrics on $M$, where $0\leqε\leq\infty$. We prove that if $M$ is hyperbolic the natural inclusion $\cT^ε(M)\hookrightarrow\cT(M)$ is, in general, not homotopically trivial. In particular, $\cT^ε(M)$ is, in general, not contractible.
We study how the existence of a negatively pinched Kähler metric on a domain in complex Euclidean space restricts the geometry of its boundary. In particular, we show that if a convex domain admits a complete Kähler metric, with pinched negative holomorphic bisectional curvature outside a compact set, then the boundary of the domain does not contain any complex subvariety of positive domain. Moreover, if the boundary of the domain is smooth, then it is of finite type in the sense of D'Angelo. We also use curvature to provide a characterization of strong pseudoconvexity amongst convex domains. …
We generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, $\mathbb{C}$) to geometrically infinite discrete isometry subgroups in the case of rank 1 symmetric spaces, and, under the assumption of bounded torsion, to the case of negatively pinched Hadamard manifolds. Every such geometrically infinite isometry subgroup $Γ$ has a set of nonconical limit points with cardinality of continuum.
We compute the $L^p$-cohomology spaces of some negatively curved manifolds. We deal with two cases: manifolds with finite volume and sufficiently pinched negative curvature, and conformally compact manifolds.
The fundamental group of a Riemannian manifold with $δ$-pinched negative curvature, $δ>1/4$, cannot be the fundamental group of a quasicompact Kähler manifold. The proof also implies that a non-uniform lattice in $F_{4(-20)}$ cannot be the fundamental group of a quasicompact Kähler manifold. We also construct examples in the spirit of Gromov-Thurston to show that our result is a non-trivial extension of the previously known result that a non-uniform lattice in real hyperbolic space in dimension at least 3 cannot be the fundamental group of a quasicompact Kähler manifold.
In this paper, we generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, $\mathbb{C}$) to geometrically infinite discrete subgroups $Γ$ of isometries of negatively pinched Hadamard manifolds $X$. We then generalize a theorem of Bishop to prove that every discrete geometrically infinite isometry subgroup $Γ$ has a set of nonconical limit points with the cardinality of the continuum.
We prove that a complete Kähler manifold with holomorphic curvature bounded between two negative constants admits a unique complete Kähler-Einstein metric. We also show this metric and the Kobayashi-Royden metric are both uniformly equivalent to the background Kähler metric. Furthermore, all three metrics are shown to be uniformly equivalent to the Bergman metric, if the complete Kähler manifold is simply-connected, with the sectional curvature bounded between two negative constants. In particular, we confirm two conjectures of R. E. Greene and H. Wu posted in 1979.
Non-existence of complete Kähler metric of negatively pinched holomorphic sectional curvature
本文证明了在有界弱伪凸域上，若Kähler度量满足特定曲率与几何条件，则其完备性决定了负向夹紧全纯截面曲率完备Kähler度量及完备Kähler-Einstein度量的存在性二分法，揭示了此类度量的不存在性充分条件。
We show that a noncompact manifold with bounded sectional curvature, whose ends are sufficiently Gromov-Hausdorff close to rays, has a finite dimensional space of square-integrable harmonic forms. In the special case of a finite-volume manifold with pinched negative sectional curvature, we show that the essential spectrum of the p-form Laplacian is the union of the essential spectra of a collection of ordinary differential operators associated to the ends. We give examples of such manifolds with curvature pinched arbitrarily close to -1 and with an infinite number of gaps in the spectrum of th…
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.
Bunching for relatively pinched metrics
本文研究曲率张量逐点负捏挤的紧致黎曼流形，改进了Hasselblatt 1994年论文中的不稳定聚束估计。
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.
A sharp vanishing theorem for the $L^p$ cohomology torsion of Riemannian manifolds with pinched negative curvature is given. It follows that certain negatively curved homogeneous spaces cannot be quasiisometric to better pinched manifolds.
Kähler manifolds with an almost $1/4$-pinched metric
本研究在Stover和Toledo发现的一类非局部对称紧Kähler流形上构造出近负1/4夹挤黎曼度量，证明其无法达到严格负1/4夹挤，且是Gromov-Thurston夹挤度量在复双曲场景的推广，填补了相关实例空白。
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.
Negative holomorphic bisectional curvature of some bounded domains
该研究证明了$\u2102^n$中边界附近具有负pinched全纯双截曲率完备Kähler度量的有界域整体也存在该度量，并推导出两类有界域满足此性质，推进了多复变几何中相关曲率问题的研究。
Manifolds with PIC1 pinched curvature
本文通过构造从任意 PIC1 夹挤的完备非紧流形出发、永恒存在的 Ricci 流，将三维非负夹挤 Ricci 曲率流形的平坦或紧性结论推广到高维。
We study the asymptotic behaviour of simply connected, Riemannian manifolds $X$ of strictly negative curvature admitting a non-uniform lattice $Γ$. If the quotient manifold $\bar X= Γ\backslash X$ is asymptotically $1/4$-pinched, we prove that $Γ$ is divergent and $U\bar X$ has finite Bowen-Margulis measure (which is then ergodic and totally conservative with respect to the geodesic flow); moreover, we show that, in this case, the volume growth of balls $B(x,R)$ in $X$ is asymptotically equivalent to a purely exponential function $c(x)e^{δR}$, where $δ$ is the topological entropy of the geodes…
Geography of pinched four-manifolds
本文研究了具有正或负夹挤截面曲率的有向四维流形，证明了其欧拉示性数和符号的新限制，并给出单连通情形下曲率介于0.161与1之间时流形同胚于S⁴或CP²的分类结果。
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…
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。
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。
We show that the hyperbolization of polyhedra pulls back regular neighborhoods of PL submanifolds. Applying this to the Riemannian version of the hyperbolization due to Ontaneda gives open complete manifolds of pinched negative curvature that are homotopy equivalent to closed smooth manifolds but contain no smooth spines. We also find open complete negatively pinched manifolds that are homotopy equivalent to closed non-smoothable manifolds.
On Almost-Fuchsian submanifodls of Hadamard spaces and the asymptotic Plateau problem
本文研究了负曲率空间中具有小曲率的极小子流形，证明了在曲率负被夹的Hadamard空间中，第二基本形受限的完备极小子流形能使渐近Plateau问题对一类无穷远球面唯一可解。
We study complete noncompact long time solutions $(M, g(t))$ to the Kähler-Ricci flow with uniformly bounded nonnegative holomorphic bisectional curvature. We will show that when the Ricci curvature is positive and uniformly pinched, i.e. $ R_\ijb \ge cRg_\ijb$ at $(p,t)$ for all $t$ for some $c>0$, then there always exists a local gradient Kähler Ricci soliton limit around $p$ after possibly rescaling $g(t)$ along some sequence $t_i \to \infty$. We will show as an immediate corollary that the injectivity radius of $g(t)$ along $t_i$ is uniformly bounded from below along $t_i$, and thus $M$ mu…
On Kanai's conjecture for frame flows over negatively curved manifolds
本文证明在严格1/4夹挤的负曲率闭流形上，若标架流遍历且相关丛光滑，则流形位似于实双曲流形，推广了Kanai的三维结果。
Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. Inspired by the theory of diophantine approximation of a real (or complex) number by rational ones, we develop a theory of approximation of geodesic lines starting from a given cusp by ones returning to it. We define a new invariant for M, the {\it Hurwitz constant} of M. It measures how well all geodesic lines starting from the cusp are approximated by ones returning to it. In the case of constant curvature, we express the Hurwitz constant in terms of lengths of closed geodesics and their dep…
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.
We show that hyperbolic 3-manifolds with finitely generated fundamental group are tame, that is the ends are products. We actually work in slightly greater generality with pinched negatively curved manifolds with hyperbolic cusps. This answers a conjecture of Marden and implies the Ahlfors measure conjecture. Applications are given to other questions about Kleinian groups and 3-manifolds.
Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. We study the asymptotics of the number of geodesics in M starting from and returning to a given cusp, and of the number of horoballs at parabolic fixed points in the universal cover of M. In the appendix, due to K. Belabas, the case of SL(2,Z) and of Bianchi groups is developed.
Let $Γ$ be a (non-elementary) convex co-compact group of isometries of a pinched Hadamard manifold $X$. We show that a normal subgroup $Γ_0$ has critical exponent equal to the critical exponent of $Γ$ if and only if $Γ/ Γ_0$ is amenable. We prove a similar result for the exponential growth rate of closed geodesics on $X / Γ$. These statements are analogues of classical results of Kesten for random walks on groups and of Brooks for the spectrum of the Laplacian on covers of Riemannian manifolds.
Let $X$ be a Hadamard manifold with pinched negative curvature $-b^2\leqκ\leq -1$. Suppose $Σ\subseteq X$ is a totally geodesic, codimension-1 submanifold and consider the geodesic flow $Φ^ν_t$ on $X$ generated by a unit normal vector field $ν$ on $Σ$. We say the normal growth exponent of $Σ$ in $X$ is at most $β$ if \[ \lim_{t \rightarrow \pm \infty} \frac{ \Vert d Φ_t^ν\Vert_{\infty} }{ e^{β\vert t \vert}} < \infty, \] where $\Vert d Φ_t^ν\Vert_{\infty} $ is the supremum of the operator norm of $d Φ_t^ν$ over all points of $Σ$. We show that if $Σ$ is bi-Lipschitz to hyperbolic $n$-space $\ma…
We consider the Laplace-Beltrami operator with Dirichlet boundary conditions on convex domains in a Riemannian manifold $(M^n,g)$, and prove that the product of the fundamental gap with the square of the diameter can be arbitrarily small whenever $M^n$ has even a single tangent plane of negative sectional curvature. In particular, the fundamental gap conjecture strongly fails for small deformations of Euclidean space which introduce any negative curvature. We also show that when the curvature is negatively pinched, it is possible to construct such domains of any diameter up to the diameter of …
Counting and equidistribution of strongly reversible closed geodesics in negative curvature
本文研究含二阶挠元的受压负曲率轨形中强可逆闭测地线，利用热力学形式体系等新技术证明其渐近计数及关于 Bowen-Margulis 测度的等分布，并给出多类新例子。
Estimates of the Poisson kernel on negatively curved Hadamard manifolds
本文研究夹挤负曲率 Hadamard 流形的 Poisson 核，借助 Anderson-Schoen 的正调和函数锥估计证明其整体上下界，并用于量化调和测度向边界点测度及无穷远边界测度的收敛。
Exponential stability for the three-dimensional Navier-Stokes equations on negatively curved manifolds
本文将双曲空间$\HH^3$上三维不可压缩Navier-Stokes方程的指数稳定性定理推广到具有有界几何（包括严格正单射半径）和负截面曲率$-b^2 \leq K \leq -a^2 < 0$的完备单连通黎曼3-流形$(M^3, g)$上，通过克服曲率非标量、Leray投影不交换和谱隙未知三个障碍，证明小$L^3$数据存在唯一全局温和解且指数衰减。
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