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

具有非负截面曲率的环面爱因斯坦4-流形

Toric Einstein 4-manifolds with nonnegative sectional curvature

本文证明了4-流形上具有非负截面曲率的$T^2$-不变爱因斯坦度量是局部对称的。

在具有非负截面曲率的流形中涉及2-张量场的Sobolev不等式

Sobolev inequalities involving 2-tensor fields in manifolds with nonnegative sectional curvature

本文应用ABP方法，在非负截面曲率流形中为光滑对称一致正定(0,2)张量场建立了Log Sobolev型及Michael Simon Sobolev不等式。

关于非负截面曲率的稳态梯度Ricci孤立子的基本群

On the fundamental group of steady gradient Ricci solitons with nonnegative sectional curvature

本文证明了具有非负截面曲率的完备稳态梯度Ricci孤立子的基本群要么平凡要么无限，并得出$n$维完备$κ$-非塌缩的此类孤立子必微分同胚于$\mathbb{R}^n$的推论。

The logarithmic Sobolev inequality for a submanifold in manifold with nonnegative sectional curvature

We prove a sharp logarithmic Sobolev inequality which holds for compact submanifolds without boundary in Riemannian manifold with nonnegative sectional curvature of arbitrary dimension and codimension, while the ambient manifold needs to have a specific Euclid-like property. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature. This extends a recent result of S. Brendle with Euclidean setting.

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.

An orbit space of a nonlinear involution of $S^2\times S^2$ with nonnegative sectional curvature

We describe a construction of Riemannian metrics of nonnegative sectional curvature on a closed smooth nonorientable 4-manifold with fundamental group of order two that realizes a homotopy class that was not previously known to contain nonnegatively curved manifolds. The procedure yields new metrics of nonnegative sectional curvature on any 2-sphere bundle with base space the 2-sphere or the real projective plane.

Geometrically formal 4-manifolds with nonnegative sectional curvature

A Riemannian manifold is called geometrically formal if the wedge product of any two harmonic forms is again harmonic. We classify geometrically formal compact 4-manifolds with nonnegative sectional curvature. If the sectional curvature is strictly positive, the manifold must be homeomorphic to S^4 or diffeomorphic to CP^2. This conclusion stills holds true if the sectional curvature is strictly positive and we relax the condition of geometric formality to the requirement that the length of harmonic 2-forms is not too nonconstant. In particular, the Hopf conjecture on S^2 x S^2 holds in this c…

On the moduli spaces of metrics with nonnegative sectional curvature

The Kreck-Stolz s invariant is used to distinguish connected components of the moduli space of positive scalar curvature metrics. We use a formula of Kreck and Stolz to calculate the s invariant for metrics on S^n bundles with nonnegative sectional curvature. We then apply it to show that the moduli spaces of metrics with nonnegative sectional curvature on certain 7-manifolds have infinitely many path components. These include the first non-homogeneous examples of this type and certain positively curved Eschenburg and Aloff-Wallach spaces.

Extrinsci radius pinching in space forms of nonnegative sectional curvature

We give new estimates for the extrinsic radius of compact hypersurfaces of the Euclidean space and the open hemisphere in terms of high order mean curvatures. Then we prove pinching results corresponding to theses estimates. We show that under a suitable pinching condition, the hypersurface is diffeomorphic and almost isometric to a geodesic hypersphere.

Nonconnected Moduli Spaces of Nonnegative Sectional Curvature Metrics on Simply Connected Manifolds

We show that in each dimension $4n+3$, $n\ge 1$, there exist infinite sequences of closed smooth simply connected manifolds $M$ of pairwise distinct homotopy type for which the moduli space of Riemannian metrics with nonnegative sectional curvature has infinitely many path components. Closed manifolds with these properties were known before only in dimension seven, and our result does also hold for moduli spaces of Riemannian metrics with positive Ricci curvature. Moreover, in conjunction with work of Belegradek, Kwasik and Schultz, we obtain that for each such $M$ the moduli space of complete…

Moduli space of metrics of nonnegative sectional or positive Ricci curvature on homotopy real projective spaces

We show that the moduli space of metrics of nonnegative sectional curvature on every homotopy ${\mathbb {R}} P^5$ has infinitely many path components. We also show that in each dimension $4k+1$ there are at least $2^{2k}$ homotopy ${\mathbb {R}} P^{4k+1}$s of pairwise distinct oriented diffeomorphism type for which the moduli space of metrics of positive Ricci curvature has infinitely many path components. Examples of closed manifolds with finite fundamental group with these properties were known before only in dimensions $4k+3\geq 7$.

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.

The moduli Space of nonnegatively curved metrics on quotients of $S^2\times S^3$ by involutions

We show that for an orientable non-spin manifold with fundamental group $\mathbb{Z}_2$ and universal cover $S^2\times S^3,$ the moduli space of metrics of nonnegative sectional curvature has infinitely many path components. The representatives of the components are quotients of the standard metric on $S^3\times S^3$ or metrics on Brieskorn varieties previously constructed using cohomogeneity one actions. The components are distinguished using the relative $η$ invariant of the spin$^c$ Dirac operator computed by means of a Lefschetz fixed point theorem.

Geometric inequalities for hypersurfaces with nonnegative sectional curvature in hyperbolic space

In this article, we will use inverse mean curvature flow to establish an optimal Sobolev-type inequality for hypersurfaces $Σ$ with nonnegative sectional curvature in $\mathbb{H}^n$. As an application, we prove the hyperbolic Alexandrov-Fenchel inequalities for hypersurfaces with nonnegative sectional curvature in $\mathbb{H}^n$: \begin{align*} \int_Σ p_{2k}\geq ω_{n-1}\left[\left(\frac{|Σ|}{ω_{n-1}}\right)^\frac{1}{k}+\left(\frac{|Σ|}{ω_{n-1}}\right)^{\frac{1}{k}\frac{n-1-2k}{n-1}}\right]^k, \end{align*} where $p_i$ is the normalized $i$-th mean curvature. Equality holds if and only if $Σ$ is…

Moduli spaces of nonnegative sectional curvature and non-unique souls

We apply various topological methods to distinguish connected components of moduli spaces of complete Riemannian metrics of nonnegative sectional curvature on open manifolds. The new geometric ingredient is that souls of nearby nonnegatively curved metrics are ambiently isotopic.

Four-dimensional Einstein manifolds with sectional curvature bounded from above

Given an Einstein structure with positive scalar curvature on a four-dimensional Riemannian manifolds, that is $Ric=λg$ for some positive constant $λ$. For convenience, the Ricci curvature is always normalized to $Ric=1$. A basic problem is to classify four-dimensional Einstein manifolds with positive or nonnegative curvature and $Ric=1$. In this paper, we firstly show that if the sectional curvature satisfies $K\le M_1= \frac{\sqrt{3}}2\approx 0.866025$, then the sectional curvature will be nonnegative. Next, we prove a family of rigidity theorems of Einstein four-manifolds with nonnegative s…

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.

Nonnegatively curved metrics on S^2xR^2

We classify the complete metrics of nonnegative sectional curvature on M^2xR^2, where M^2 is any compact 2-manifold.

正曲率真空静态空间的刚性

Rigidity of positively curved vacuum Static spaces

本文研究维数≥4且具非负截面曲率的紧致真空静态空间，证明正截面曲率且Weyl曲率张量完全散度非负的这类空间等距于标准球面，还推广了临界点方程相关的刚性结果。

A PL-manifold of nonnegative curvature homeomorphic to $S^2 \times S^2$ is a direct metric product (Preliminary version)

Let $M^4$ be a PL-manifold of nonnegative curvature that is homeomorphic to a product of two spheres, $S^2 \times S^2$. We prove that $M$ is a direct metric product of two spheres endowed with some polyhedral metrics. In other words, $M$ is a direct metric product of the surfaces of two convex polyhedra in $\mathbb{R}^3$. The background for the question is the following. The classical H.Hopf's hypothesis states: for any Riemannian metric on $S^2 \times S^2$ of nonnegative sectional curvature the curvature cannot be strictly positive at all points. There is no quick answer to this question: it …

具有非负曲率流形中的子流形直径估计

Diameter estimates for submanifolds in manifolds with nonnegative curvature

研究了闭合连通流形在非负截面曲率流形中的内在直径估计，基于均曲率场积分，并扩展到带边的紧凸表面。

关于具有非负全纯截面曲率的紧凯勒流形的结构

On the structure of compact Kähler manifolds with nonnegative holomorphic sectional curvature

本文研究具有非负全纯截面曲率的紧凯勒流形，通过建立完整和原理的伪有效版本，证明在无真正平坦切向量等条件下流形必为射影且有理连通，从而回答了 Matsumura-Yang 问题并推广了 Yau 猜想，同时分类了非射影三维情形并推广到正实双截面曲率度量。

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.

Ricci flow and nonnegativity of curvature

In this paper, we prove a general maximum principle for the time dependent Lichnerowicz heat equation on symmetric tensors coupled with the Ricci flow on complete Riemannian manifolds. As an application we construct complete manifolds with bounded nonnegative sectional curvature of dimension greater than or equal to four such that the Ricci flow does not preserve the nonnegativity of the sectional curvature, even though the nonnegativity of the sectional curvature was proved to be preserved by Hamilton in dimension three. The example is the first of this type. This fact is proved through a gen…

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.

$CP^3$ 及其 blow-up 上向量丛的非负曲率

Nonnegative curvature on vector bundles over ${CP}^3$ and its blow-up

本文证明 $CP^3$ 上秩至少七、$CP^3\#CP^3$ 上秩至少八的实向量丛及其单位球面丛具有非负截面曲率的完备度量，通过低维丛分类与 Grove--Ziller 提升理论实现。

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

Index theory and deformations of open nonnegatively curved manifolds

We use an index-theoretic technique of Hitchin to show that the space of complete Riemannian metrics of nonnegative sectional curvature on certain open spin manifolds has nontrivial homotopy groups in infinitely many degrees. A new ingredient of independent interest is homotopy density of the subspace of metrics with cylindrical ends.

Real affine varieties of nonnegative curvature

Let $X_{\mathbb{C}}$ be a smooth real affine variety with compact real points $X_{\mathbb{R}}$. We show that $X_{\mathbb{C}}$ is diffeomorphic to the normal bundle of $X_{\mathbb{R}}$ provided that $X_{\mathbb{C}}$ admits a complete Riemannian metric of nonnegative sectional curvature which is also invariant under the conjugation.

Rigidity for Nonnegatively Curved Metrics on S^2xR^3

We address the question: how large is the family of complete metrics with nonnegative sectional curvature on S^2xR^3? We classify the connection metrics, and give several examples of non-connection metrics. We provide evidence that the family is small by proving some rigidity results for metrics more general than connection metrics.

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.

具有非负中间Ricci曲率的流形中的Sobolev不等式

Sobolev inequalities in manifolds with nonnegative intermediate Ricci curvature

本文利用Alexandrov-Bakelman-Pucci方法，将Michael-Simon型Sobolev不等式推广到具有非负中间Ricci曲率的Riemannian流形中的子流形上，并由此导出极小子流形的等周不等式。

Three Circle Theorems on Kähler manifolds and applications

The classical Hadamard three circle theorem is generalized to complete Kähler manifolds. More precisely, we show that the nonnegativity of the holomorphic sectional curvature is a necessary and sufficient condition for the three circle theorem. As corollaries, two sharp monotonicity formulae for holomorphic functions are derived. Among applications, we derive sharp dimension estimates (with rigidity) of holomorphic functions with polynomial growth when the holomorphic sectional curvature is nonnegative. When the bisectional curvature is nonnegative, this was due to Ni. Also we study holomorphi…

A note on K$\ddot{a}$hler manifolds with almost nonnegative bisectional curvature

In this note we prove the following result: There is a positive constant $ε(n,Λ)$ such that if $M^n$ is a simply connected compact K$\ddot{a}$hler manifold with sectional curvature bounded from above by $Λ$, diameter bounded from above by 1, and with holomorphic bisectional curvature $H \geq -ε(n,Λ)$, then $M^n$ is diffeomorphic to the product $M_1\times ... \times M_k$, where each $M_i$ is either a complex projective space or an irreducible K$\ddot{a}$hler-Hermitian symmetric space of rank $\geq 2$. This resolves a conjecture of F. Fang under the additional upper bound restrictions on section…

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.

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。

关于非紧 Hessian 流形的拓扑刚性定理

A topological rigidity theorem on noncompact Hessian manifolds

本文通过在非紧仿射黎曼流形上构造几何流的短时间解，建立了具有非负 Hessian 截曲率的 Hessian 度量，并证明了切丛具有最大体积增长的完备非紧 Hessian 流形微分同胚于 $\mathbb{R}^n$。

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.

截面曲率与矩阵位移凸性

Sectional curvature and matrix displacement convexity

该研究证明黎曼流形截面曲率非负等价于熵张量的矩阵位移凸性，还将此结论应用于强化HWI不等式等结果，且内在维数Wasserstein收缩不等式可刻画非负截面曲率

Morse-Novikov cohomology of almost nonnegatively curved manifolds

Let $M^n$ be a closed manifold of almost nonnegative sectional curvature and nonzero first de Rham cohomology group. For any $[θ] \in H^1_{dR}(M^n), [θ] \neq 0$, we show that the Morse- Novikov cohomology group $H^p(M^n, θ)$ vanishes for any $p$. A similar result holds for a closed manifold of almost nonnegative Ricci curvature under the additional assumption that its curvature operator is uniformly bounded from below.

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.

非负曲率局部对称空间中椭圆与抛物方程的凹性

Concavity for elliptic and parabolic equations in locally symmetric spaces with nonnegative curvature

本研究将Langford和Scheuer的结果拓展，建立了非负截面曲率局部对称空间上椭圆与抛物方程解的凹性原理，是首个非恒定截面曲率空间上的通用凹性原理。

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 $b_j$ ($2\le j\le I-2$).

Classification of Cohomogeneity One Manifolds in Low Dimensions

A cohomogeneity one manifold is a manifold with the action of a compact Lie group, whose quotient is one dimensional. Such manifolds are of interest in Riemannian geometry, in the context of nonnegative sectional curvature, as well as in other areas of geometry and in physics. In this paper we classify compact simply connected cohomogeneity one manifolds in dimensions 5, 6 and 7. We also show that all such manifolds admit metrics of nonnegative sectional curvature, with the possible exception of two families of manifolds.

The weighted mixed curvature of a foliated manifold

In this paper, we introduce the weighted mixed (sectional, Ricci and scalar) curvature of a foliated (and almost-product) Riemannian manifold $(M,g)$ equipped with a vector field $X$. We define several functions ($q$th Ricci type curvatures), which "interpolate" between the weighed sectional and Ricci curvatures. The novel concepts of the "mixed curvature-dimension" condition and "synthetic dimension of a distribution" allow us to update the estimate of the diameter of a compact Riemannian foliation and to prove new splitting theorems for almost-product manifolds of nonnegative/nonpositive wei…

On the structure of complete 3-manifolds with nonnegative scalar curvature

In this paper we will show the following result: Let $\mathcal{N} $ be a complete (noncompact) connected orientable Riemannian three-manifold with nonnegative scalar curvature $S \geq 0$ and bounded sectional curvature $ K_{s} \leq K $. Suposse that $Σ\subset \mathcal{N} $ is a complete orientable connected area-minimizing cylinder so that $π_1 (Σ) \in π_1 (\mathcal{N})$. Then $\mathcal{N}$ is locally isometric either to $\mathbb{S} ^1 \times \mathbb{R} ^2 $ or $\mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{R}$ (with the standard product metric). As a corollary, we will obtain: Let $\mathcal…

Nonnegative curvature on stable bundles over compact rank one symmetric spaces

In this note we show that every (real or complex) vector bundle over a compact rank one symmetric space carries, after taking the Whitney sum with a trivial bundle of sufficiently large rank, a metric with nonnegative sectional curvature. We also examine the case of complex vector bundles over other manifolds, and give upper bounds for the rank of the trivial bundle that is necessary to add when the base is a sphere.

Diffeomorphic souls and disconnected moduli spaces of nonnegatively curved metrics

We give examples of open manifolds that carry infinitely many complete metrics of nonnegative sectional curvature such that they all have the same soul, and their isometry classes lie in different connected components of the moduli space. All previously known examples of this kind have souls of codimension one. In our examples the souls have codimensions three and two.

On the moduli space of nonnegatively curved metrics on Milnor spheres

Let $M$ be a Milnor sphere or, more generally, the total space of a linear $S^3$-bundle over $S^4$ with $H^4(M;\mathbb{Q})=0$. We show that the moduli space of metrics of nonnegative sectional curvature on $M$ has infinitely many path components. The same holds true for the moduli space of metrics of positive Ricci curvature on $M$.

Classification of gradient steady Ricci solitons with linear curvature decay

In this paper, we give a description for steady Ricci solitons with a linear decay of sectional curvature. In particular, we classify all 3-dimensional steady Ricci solitons and 4-dimensional $κ$-noncollpased steady Ricci solitons with nonnegative sectional curvature under the linear curvature decay.

Topological obstructions to nonnegative curvature

We find new obstructions to the existence of complete Riemannian metric of nonnegative sectional curvature on manifolds with infinite fundamental groups. In particular, we construct many examples of vector bundles whose total spaces admit no nonnegatively curved metric.

Kervaire 九球面上的正截面曲率

Positive sectional curvature on homotopy nine-spheres

利用相容圆盘定理构造附着映射，在 Kervaire 九球面上获得正截面曲率，并通过 Dirac 指标排除部分类型的非负曲率。

三维奇异 Ricci 流的非负 Ricci 曲率极限的伪局部性与完备性

Pseudolocality and completeness for nonnegative Ricci curvature limits of 3D singular Ricci flows

本文研究了从三维完备非紧 Riemann 流形产生的奇异 Ricci 流的完备性，通过结合伪局部性结果证明在特定体积比下界或紧支撑扰动条件下该流在正时刻是完备的。

Properly discontinuous actions on Hilbert manifolds

In this article we study properly discontinuous actions on Hilbert manifolds giving new examples of complete Hilbert manifolds with nonnegative, respectively nonpositive, sectional curvature with infinite fundamental group. We also get examples of complete infinite dimensional Kähler manifolds with positive holomorphic sectional curvature and infinite fundamental group in contrast with the finite dimensional case and we classify abelian groups acting linearly, isometrically and properly discontinuously on Stiefel manifolds. Finally, we classify homogeneous Hilbert manifolds with constant secti…