Invariant metrics with nonnegative curvature on compact Lie groups

Invariant metrics with nonnegative curvature on SO(4) and other Lie groups

We develop techniques for classifying the nonnegatively curved left-invariant metrics on a compact Lie group G. We prove rigidity theorems for general G and a partial classification for G=SO(4). Our approach is to reduce the general question to an infinitesimal version; namely, to classify the directions one can move away from a fixed bi-invariant metric such that curvature variation formulas predict nearby metrics are nonnnegatively curved.

非负Ricci曲率与一致凸边界迫使紧致性

Nonnegative Ricci Curvature and Uniformly Convex Boundary Forces Compactness

研究具有非负Ricci曲率和一致凸边界的完备黎曼流形的紧致性，通过构造Neumann条件下正调和函数的单调量证明其紧致且基本群有限。

Techniques for classifying nonnegatively curved left-invariant metrics on compact Lie groups

We provide techniques for studying the nonnegatively curved left-invariant metrics on a compact Lie group. For "straight" paths of left-invariant metrics starting at bi-invariant metrics and ending at nonnegatively curved metrics, we deduce a nonnegativity property of the initial derivative of curvature. We apply this result to obtain a partial classification of the nonnegatively curved left-invariant metrics on SO(4).

The codimension one homology of a complete manifold with nonnegative Ricci curvature

In this paper we prove that a complete noncompact manifold with nonnegative Ricci curvature has a trivial codimension one homology unless it is a split or flat normal bundle over a compact totally geodesic submanifold. In particular, we prove the conjecture that a complete noncompact manifold with positive Ricci curvature has a trivial codimension one integer homology. We also have a corollary stating when the codimension two integer homology of such a manifold is torsion free.

具有非负Ricci曲率和欧几里得体积增长的完备4-流形的大尺度结构

The large scale structure of complete $4$-manifolds with nonnegative Ricci curvature and Euclidean volume growth

该研究综述了与合作者关于非负Ricci曲率、欧几里得体积增长的完备4-流形爆破结构的工作，指出此类流形无穷远处近似球面空间形式上的锥，并提出开放问题及未来研究方向。

非负数量曲率下向极端极限空间的几何收敛

Geometric Convergence to an Extreme Limit Space with nonnegative scalar curvature

本文研究非负数量曲率流形序列收敛到极端极限空间的问题，证明了序列在保体积内蕴平坦和测度格罗莫夫-豪斯多夫意义下收敛，为检验广义非负数量曲率定义提供了测试空间。

具有非负双正交曲率的近凯勒四维流形

Almost-Kähler four manifolds with nonnegative biorthogonal curvature

本文对具有非负双正交曲率的紧致近凯勒四维流形进行了完整分类。

Compact 4-manifolds with harmonic Weyl tensor and nonnegative biorthogonal curvature

In this note we classify compact 4-manifolds with harmonic Weyl tensor and nonnegative biorthogonal curvature

Manifolds with almost nonnegative curvature operator and principal bundles

We study manifolds with almost nonnegative curvature operator (ANCO) and provide first examples of closed simply connected ANCO mannifolds that do not admit nonnegative curvature operator.

On shrinking Gradient Ricci Soliton With Nonnegative Sectional Curvature

In this paper, we prove some rigidity theorems for shrinking gradient Ricci solitons with nonnegative sectional curvature.

Horizontally homothetic submersions and nonnegative curvature

We show that any horizontally homothetic submersion from a compact manifold of nonnegative sectional curvature is a Riemannian submersion.

Manifolds of positive Ricci curvature, quadratically asymptotically nonnegative curvature, and infnite Betti numbers

In a previous paper, we constructed complete manifolds of positive Ricci curvature with quadratically asymptotically nonnegative curvature and infinite topological type but dimension $\ge 6$. The purpose of the present paper is to use a different way to exhibit a family of complete $I$-dimensinal ($I\ge5$) Riemannian manifolds of positive Ricci curvature, quadratically asymptotically nonnegative sectional curvature, and certain infinite Betti number $b_j$ ($2\le j\le I-2$).

Sobolev inequalities in manifolds with nonnegative curvature

We prove a sharp Sobolev inequality on manifolds with nonnegative Ricci curvature. Moreover, we prove a Michael-Simon inequality for submanifolds in manifolds with nonnegative sectional curvature. Both inequalities depend on the asymptotic volume ratio of the ambient manifold.

A duality theorem for Riemannian foliations in nonnegative sectional curvature

Using a new type of Jacobi field estimate we will prove a duality theorem for singular Riemannian foliations in complete manifolds of nonnegative sectional curvature.

具有非负截面曲率的四元Kähler流形

Quaternionic-Kähler manifolds with nonnegative sectional curvature

本文证明具有正数量曲率和非负截面曲率的紧四元Kähler流形等距于对称空间，推广了Berger的经典定理。

Sobolev inequalities on manifolds with nonnegative Bakry-Émery Ricci curvature

In this note we extend a recent result of S. Brendle [3] to Riemannian manifolds with densities and nonnegative Bakry-Émery Ricci curvature.

Nonnegative curvature, symmetry and fundamental group

We prove a result on equivariant deformations of flat bundles, and as a corollary, we obtain two ``splitting in a finite cover'' theorems for isometric group actions on Riemannian manifolds with infinite fundamental groups, where the manifolds are either compact of nonnegative Ricci curvature, or complete of nonnegative sectional curvature.

An example of compact Kähler manifold with nonnegative quadratic bisectional curvature

We construct a compact Kähler manifold of nonnegative quadratic bisectional curvature, which does not admit any Kähler metric of nonnegative orthogonal bisectional curvature. The manifold is a 7-dimensional Kähler C-space with second Betti number equal to 1, and its canonical metric is a Kähler-Einstein metric of positive scalar curvature

The Fundamental Groups of Open Manifolds with Nonnegative Ricci Curvature

We survey the results on fundamental groups of open manifolds with nonnegative Ricci curvature. We also present some open questions on this topic.

On 3-manifolds with pointwise pinched nonnegative Ricci curvature

There is a conjecture that a complete Riemannian 3-manifold with bounded sectional curvature, and pointwise pinched nonnegative Ricci curvature, must be flat or compact. We show that this is true when the negative part (if any) of the sectional curvature decays quadratically.

Nonnegative curvature, elliptic genus and unbounded Pontryagin numbers

We discuss the cobordism type of spin manifolds with nonnegative sectional curvature. We show that in each dimension $4k \geq 12$, there are infinitely many cobordism types of simply connected and nonnegatively curved spin manifolds. Moreover, we raise and analyze a question about possible cobordism obstructions to nonnegative curvature.

Maximal first Betti number rigidity for open manifolds of nonnegative Ricci curvature

Let $M$ be an open Riemannian $n$-manifold with nonnegative Ricci curvature. We prove that if the first Betti number of $M$ equals $n-1$, then $M$ is flat.

Open manifolds with asymptotically nonnegative Ricci curvature and large volume growth

In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature and large volume growth. We prove that they have finite topological types under some curvature decay and volume growth conditions. We also generize it to the manifolds with $k$-th asymptotically nonnegative Ricci curvature by using extensions of Abresch-Gromoll's excess function estimate.

Einstein manifolds with nonnegative isotropic curvature are locally symmetric

Let (M,g) be an Einstein manifold of dimension n \geq 4 with nonnegative isotropic curvature. We show that (M,g) is locally symmetric.

Seven dimensional cohomogeneity one manifolds with nonnegative curvature

We show that a certain family of cohomogeneity one manifolds does not admit an invariant metric of nonnegative sectional curvature, unless it admits one with positive curvature. As a consequence, the classification of nonnegatively curved cohomogeneity one manifolds in dimension 7 is reduced to only one further family of candidates

Almost Nonnegative Ricci curvature and new vanishing theorems for genera

We derive several vanishing theorems for genera under almost nonnegative Ricci curvature and infinite fundamental group, which includes Todd genus, $\widehat{A}$-genus, elliptic genera and Witten genus. A vanishing theorem of Euler characteristic number for almost nonnegatively curved Alexandrov spaces is also proved.

Gap theorem on Kähler manifold with nonnegative orthogonal bisectional curvature

In this paper we prove a gap theorem for Kähler manifolds with nonnegative orthogonal bisectional curvature and nonnegative Ricci curvature, which generalizes an earlier result of the first author. We also prove a Liouville theorem for plurisubharmonic functions on such a manifolds, which generalizes a previous result of L.-F. Tam and the first author.

具有非负 Ricci 曲率的完备 Kähler 流形 II

Complete Kähler manifolds with nonnegative Ricci curvature II

本文研究了具有非负 Ricci 曲率的 Kähler 流形上的刚性，并证明满足特定曲率、体积增长与曲率衰减条件的完备非紧 Kähler 曲面双全纯于仿射代数簇的解消。

The Splitting Theorem and Topology of Noncompact Spaces with Nonnegative N-Bakry Émery Ricci Curvature

In this paper, we generalize topological results known for noncompact manifolds with nonnegative Ricci curvature to spaces with nonnegative $N$-Bakry Émery Ricci curvature. We study the Splitting Theorem and a property called the geodesic loops to infinity property in relation to spaces with nonnegative $N$-Bakry Émery Ricci Curvature. In addition, we show that if $M^n$ is a complete, noncompact Riemannian manifold with nonnegative $N$-Bakry Émery Ricci curvature where $N>n$, then $H_{n-1}(M,\mathbb{Z})$ is $0$.

The isoperimetric inequality on asymptotically flat manifolds with nonnegative scalar curvature

In this note, we consider the isoperimetric inequality on an asymptotically flat manifold with nonnegative scalar curvature, and improve it by using Hawking mass. We also obtain a rigidity result when equality holds for the classical isoperimetric inequality on an asymptotically flat manifold with nonnegative scalar curvature.

Topology of Manifolds with Asymptotically Nonnegative Ricci Curvature

In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature. We show that a complete noncompact manifold with asymptoticaly nonnegative Ricci curvature and sectional curvature decay at most quadratically is diffeomorphic to a Euclidean n-space R^n under some conditions on the density of rays starting from the base point p or on the volume growth of geodesic balls in M.

Form-type Calabi-Yau equations on Kähler manifolds of nonnegative orthogonal bisectional curvature

In this paper we prove the existence and uniqueness of the form-type Calabi-Yau equation on Kähler manifolds of nonnegative orthogonal bisectional curvature.

Moduli space of metrics of nonnegative sectional or positive Ricci curvature on Brieskorn quotients in dimension 5

We show that the moduli space of Riemannian metrics of nonnegative sectional curvature on certain quotients of Brieskorn varieties in dimension 5 has infinitely many path components. The same is true for the corresponding moduli space of positive Ricci curvature.

3-manifolds with nonnegative Ricci curvature

For a noncompact 3-manifold with nonnegative Ricci curvature, we prove that either it is diffeomorphic to $\mathbb{R}^3$ or the universal cover splits. As a corollary, it confirms a conjecture of Milnor in dimension 3.

Sobolev Inequalities In Manifolds With Asymptotically Nonnegative Curvature

Using the ABP-method as in a recent work by Brendle, we establish some sharp Sobolev and isoperimetric inequalities for compact domains and submanifolds in a complete Riemannian manifold with asymptotically nonnegative curvature. These inequalities generalize those given by Brendle in the case of complete Riemannian manifolds with nonnegative curvature.

Almost Nonnegative Curvature on Some Fake $RP^{6}s$ and $RP^{14}s$

We apply the lifting theorem of Searle and the second author to put metrics of almost nonnegative curvature on the fake RP^{6}s of Hirsch and Milnor and on the analogous fake RP^{14}s.

非负曲率的开Alexandrov空间

Open Alexandrov spaces of nonnegative curvature

本文研究具有灵魂的非负曲率开Alexandrov空间，建立了与开黎曼流形非负截面曲率情形相对应的若干结构性结果。

On the existence of isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth

In this paper we provide new existence results for isoperimetric sets of large volume in Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth. We find sufficient conditions for their existence in terms of the geometry at infinity of the manifold. As a byproduct we show that isoperimetric sets of big volume always exist on manifolds with nonnegative sectional curvature and Euclidean volume growth. Our method combines an asymptotic mass decomposition result for minimizing sequences, a sharp isoperimetric inequality on nonsmooth spaces, and the concavity property of t…

Kähler-Ricci Shrinkers and Ancient Solutions with Nonnegative Orthogonal Bisectional Curvature

In this paper we prove classification results for gradient shrinking Ricci solitons under two invariant conditions, namely nonnegative orthogonal bisectional curvature and weakly PIC1, without any curvature bound. New results on ancient solutions for the Ricci and Kähler-Ricci flow are also obtained. The main new feature is that no curvature upper bound is assumed.

Remark on an inequality for closed hypersurfaces in complete manifolds with nonnegative Ricci curvature

We give a simple proof of a recent result due to Agostiniani, Fogagnolo and Mazzieri.

Conditions for Nonnegative Curvature on Vector Bundles and Sphere Bundles

This paper addresses Cheeger and Gromoll's question of which vector bundles admit a complete metric of nonnegative curvature, and relates their question to the issue of which sphere bundles admit a metric of positive curvature. We show that any vector bundle which admits a metric of nonnegative curvature must admit a connection, a tensor, and a metric on the base space which together satisfy a certain differential inequality. On the other hand, a slight sharpening of this condition is sufficient for the associated sphere bundle to admit a metric of positive curvature. Our results sharpen and g…

Graphs with nonnegative curvature outside a finite subset, harmonic functions and number of ends

We study graphs with nonnegative Bakry-Émery curvature or Ollivier curvature outside a finite subset. For such a graph, via introducing the discrete Gromov-Hausdorff convergence we prove that the space of bounded harmonic functions is finite dimensional, and as a corollary the number of non-parabolic ends is finite.

A note on nonnegative curvature and (rational) homotopy type

We discuss a question by Felix, Oprea, and Tanre concerning nonnegative curvature and (rational) homotopy type.

非负曲率黎曼流形上的sharp p-塞伯利格不等式极值函数的广义存在性

Generalized existence of extremizers for the sharp $p$-Sobolev inequality on Riemannian manifolds with nonnegative curvature

研究非负曲率黎曼流形上sharp p-塞伯利格不等式极值函数的广义存在性，通过非光滑方法和Cheeger能量收敛性分析，揭示极值函数的全局行为。

Four-dimensional cohomogeneity one Ricci flow and nonnegative sectional curvature

We exhibit the first examples of closed 4-manifolds with nonnegative sectional curvature that lose this property when evolved via Ricci flow.

非负瑞奇曲率空间的新拓扑限制

New Topological Restrictions For Spaces With Nonnegative Ricci Curvature

该研究通过贝蒂数刚性定理和简化体积消失定理，为非负瑞奇曲率空间提供了新的拓扑限制，并扩展了G. Liu的非紧致3流形分类证明。

The Weyl problem with nonnegative Gauss curvature in hyperbolic space

In this paper, we discuss the isometric embedding problem in hyperbolic space with nonnegative extrinsic curvature. We prove a priori bounds for the trace of the second fundamental form H and extend the result to n-dimensions. We also obtain an estimate for the gradient of the smaller principal curvature in 2 dimensions.

4-dimensional compact manifolds with nonnegative biorthogonal curvature

The goal of this article is to study the pinching problem proposed by S.-T. Yau in 1990 replacing sectional curvature by one weaker condition on biorthogonal curvature. Moreover, we classify 4-dimensional compact oriented Riemannian manifolds with nonnegative biorthogonal curvature. In particular, we obtain a partial answer to Yau Conjecture on pinching theorem for 4-dimensional compact manifolds.

Twisted submersions in nonnegative sectional curvature

B. Wilking introduced the dual foliation associated to a metric foliation in a Riemannian manifold with nonnegative sectional curvature, and proved that when the curvature is strictly positive, the dual foliation contains a single leaf, so that any two points in the ambient space can be joined by a horizontal curve. We show that the same phenomenon often occurs for nonnegatively curved Riemannian submersions even without the strict positive curvature condition, and irrespective of the particular metric.

具有渐近非负Bakry-Émery Ricci曲率的流形上的Sobolev不等式

Sobolev Inequality on Manifolds With Asymptotically Nonnegative Bakry-Émery Ricci Curvature

本文受已有工作启发，在具有密度且Bakry-Émery Ricci曲率渐近非负的流形上证明了Sobolev不等式。

The smoothness of Riemannian submersions with nonnegative sectional curvature

Let $M^n$ be a complete, non-compact and $C^\infty$-smooth Riemannian manifold with nonnegative sectional curvature. Suppose $\Cal S$ is a soul of $M^n$. Then any distance non-increasing retraction $Ψ: M^n \to \Cal S$ must give rise to a $C^\infty$-smooth Riemannian submersion.

闭合流形的欧拉特征与近非负曲率算子

Euler Characteristic of Closed Manifolds with Almost Nonnegative Curvature Operator

研究具有近非负曲率算子的闭合流形，解决Herrmann-Sebastian-Tuschmann关于其欧拉特征符号的问题，证明在特定条件下欧拉特征非负，并在无限基本群条件下证明欧拉特征、签名及$\widehat{A}$-特征 vanish。

Strongly nonnegative curvature

We prove that all currently known examples of manifolds with nonnegative sectional curvature satisfy a stronger condition: their curvature operator can be modified with a 4-form to become positive-semidefinite.

The set of vertices with positive curvature in a planar graph with nonnegative curvature

In this paper, we give the sharp upper bound for the number of vertices with positive curvature in a planar graph with nonnegative combinatorial curvature. Based on this, we show that the automorphism group of a planar---possibly infinite---graph with nonnegative combinatorial curvature and positive total curvature is a finite group, and give an upper bound estimate for the order of the group.