Jiayin Pan
We survey the results on fundamental groups of open manifolds with nonnegative Ricci curvature. We also present some open questions on this topic.
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In this paper, we prove some rigidity theorems for shrinking gradient Ricci solitons with nonnegative sectional curvature.
Complete non-compact gradient Ricci solitons with nonnegative Ricci curvature
本文给出了具有非负里奇曲率的完全非紧扩张（或定常）梯度Ricci孤子的数量曲率的时滞估计。作为应用，我们证明了任何具有正压缩里奇曲率的完全非紧膨胀（或定常）梯度Kähler-Ricci孤子都是Ricci平坦的.这一结果回答了Chow，Lu和Ni在一本书中提出的Kähler-Ricci孤子的问题。
In this note we extend a recent result of S. Brendle [3] to Riemannian manifolds with densities and nonnegative Bakry-Émery 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.
Stable weighted minimal surfaces in manifolds with nonnegative Bakry-Emery Ricci tensor
本文研究了流形中具有非负Bakry-Emery里奇曲率的稳定加权极小超曲面.我们将给出一些几何和拓扑应用。特别地，在$f$有界的假设下，给出了具有非负Bakry-Emery里奇曲率的完备三维流形的部分分类.
Ancient Ricci flows with nonnegative Ricci curvature
本文研究了非负曲率非塌陷古老里奇流的渐近几何，通过其无穷远处的非塌陷F-极限度量孤子得出两个分歧定理。
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 $0$.
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.
3-manifolds with nonnegative Ricci curvature
对于具有非负里奇曲率的非紧三维流形，我们证明了它要么与$\mathbb{R}^3$是同构的，要么是泛覆盖分裂的.作为一个推论，它证实了米尔诺在三维猜想。
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.
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.
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。
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.
Topology of Manifolds with Asymptotically Nonnegative Ricci Curvature
本文研究了具有渐近非负里奇曲率的完备非紧黎曼流形的拓扑。证明了具有渐近非负里奇曲率且截面曲率至多二次衰减的完备非紧流形在一定条件下与欧氏n-空间R^n是同构的，条件是从基点p出发的射线的密度或M中测地球的体积增长.
We give a simple proof of a recent result due to Agostiniani, Fogagnolo and Mazzieri.
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.
We exhibit the first examples of closed 4-manifolds with nonnegative sectional curvature that lose this property when evolved via Ricci flow.
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 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。
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。
Let $M^n$ be a compact K$\ddot{a}$hler manifold with almost nonnegative Ricci curvature and nonzero first Betti number. We show that the holomorphic Euler number of $M^n$ vanishes, which gives a new obstruction for compact complex manifolds admitting K$\ddot{a}$hler metrics with almost nonnegative Ricci curvature. A crucial step in the proof is to show a vanishing theorem of Dolbeault-Morse-Novikov cohomology.
In this paper, we construct a complete n-dim Riemannian manifold with positive Ricci curvature, quadratically nonnegatively curved infinity and infinite topological type. This gives a negative answer to a conjecture by Jiping Sha and Zhongmin Shen in the case of n greater than or equal to 6.
We establish a min-max estimate on the volume width of a closed Riemannian manifold with nonnegative Ricci curvature. More precisely, we show that every closed Riemannian manifold with nonnegative Ricci curvature admits a PL Morse function whose level set volume is bounded in terms of the volume of the manifold. As a consequence of this sweep-out estimate, there exists an embedded, closed (possibly singular) minimal hypersurface whose volume is bounded in terms of the volume of the manifold.
Optimal volume bound and volume growth for Ricci-nonnegative manifolds with positive Bi-Ricci curvature
本文在维数小于八的条件下，证明了具有非负Ricci曲率且紧集外双Ricci曲率下界为n-2的完备黎曼流形的最优体积增长，回答了Antonelli-Xu的问题，并给出正双Ricci曲率下Gromov体积界猜想的类比。
In this article, we prove Talenti's comparison theorem for Poisson equation on complete noncompact Riemannian manifold with nonnegative Ricci curvature. Furthermore, we obtain the Faber-Krahn inequality for the first eigenvalue of Dirichlet Laplacian, $L^1$- and $L^\infty$-moment spectrum, especially Saint-Venant theorem for torsional rigidity and a reverse Hölder inequality for eigenfunctions of Dirichlet Laplacian.
In this paper, we prove that any complete shrinking gradient Kähler-Ricci solitons with positive orthogonal bisectional curvature must be compact. We also obtain a classification of the complete shrinking gradient Kähler-Ricci solitons with nonnegative orthogonal bisectional curvature.
Harmonic Functions on Manifolds with Nonnegative Ricci Curvature and Linear Volume Growth
里奇曲率的下界限制了集合的体积和黎曼流形上调和函数的存在性。1975年，Shing Tung Yau证明了具有非负里奇曲率的完备非紧流形不存在次线性增长的非常数调和函数。在同一篇论文中，丘利用这一结果证明了具有非负里奇曲率的完备非紧流形至少具有线性体积增长。本文证明了关于这些流形上的调和函数的下列定理。定理：设M是具有非负里奇曲率和至多线性体积增长的完备非紧流形.如果存在一个非常数的调和函数f，它是任意给定次数q的多项式增长，则流形等距分裂，M= N x R。
On the global structure of conformal gradient solitons with nonnegative Ricci tensor
本文证明了任何具有非负Ricci张量的完备共形梯度孤子要么等距于直积$\mathbb{R}\times N^{n-1}$，要么整体共形等价于欧几里得空间$\mathbb{R}^{n}$或球面$\mathbb{S}^{n}$ .特别地，我们证明了任何完整的，非紧的，梯度Yamabe型孤子与正Ricci张量是旋转对称的，只要势函数是非常数。
$4d$ steady gradient Ricci solitons with nonnegative curvature away from a compact set
本文研究紧集外非负曲率的4维非紧κ-非塌缩稳态梯度Ricci孤立子，证明若其重标化流收敛于收缩商柱面族，则缩放后等价于Bryant Ricci孤立子，推进了该类孤立子的渐近分类研究。
Conformally flat manifolds with nonnegative Ricci curvature
我们证明了具有非负里奇曲率的n>2维完备共形平坦流形具有很好的刚性性质：它们要么是平坦的，要么局部等距于球面和直线的乘积，要么整体共形等价于平坦空间或球面空间形式.这扩展了Q. M. Cheng，M.H. Noronha，B. L. Chen和X.- P. Zhu和S.竹
Based on properties of n-subharmonic functions we show that a complete, noncompact, properly embedded hypersurface with nonnegative Ricci curvature in hyperbolic space has an asymptotic boundary at infinity of at most two points. Moreover, the presence of two points in the asymptotic boundary is a rigidity condition that forces the hypersurface to be an equidistant hypersurface about a geodesic line in hyperbolic space. This gives an affirmative answer to the question raised by Alexander and Currier in 1990.
The Topology of Open Manifolds with Nonnegative Ricci Curvature
我们调查所有结果的拓扑结构完整的非紧黎曼流形与非负里奇曲率，没有额外的条件以外的限制的尺寸，体积增长或直径增长的流形。我们还将提出相关的例子，并列出悬而未决的问题。
We study the fundamental group of an open $n$-manifold $M$ of nonnegative Ricci curvature. We show that if there is an integer $k$ such that any tangent cone at infinity of the Riemannian universal cover of $M$ is a metric cone, whose maximal Euclidean factor has dimension $k$, then $π_1(M)$ is finitely generated. In particular, this confirms the Milnor conjecture for a manifold whose universal cover has Euclidean volume growth and the unique tangent cone at infinity.
We consider Finsler submanifolds $M^n$ of nonnegative Ricci curvature in a Minkowski space $\mathbb{M}^{n+p}$ which contain a line or whose relative nullity index is positive. For hypersurfaces, submanifolds of codimension two or of dimension two, we prove that the submanifold is a cylinder, under a certain condition on the inertia of the pencil of the second fundamental forms. We give an example of a surface of positive flag curvature in a three-dimensional Minkowski space which is not locally convex.
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。
Noncompact Manifolds with Nonnegative Ricci Curvature
设（M，d）是度量空间.对于0<r<R，且p在M中，设G（p，r，R）是通过考虑所有以p为基础的、其像包含在半径为r的闭球中的回路而得到的群，并且如果它们之间存在包含在半径为R的开球中的同伦，则识别两个回路。本文研究了具有非负里奇曲率的完备开流形G（p，r，R）群的渐近性态.我们还发现了G（p，r，R）群与度量空间无穷远切锥之间的关系，并证明了具有非负里奇曲率和小线性直径增长的完备开流形的无穷远切锥是它自己的泛覆盖.
Noncoercive Ricci flow invariant curvature cones
本文研究了里奇流保持曲率的非负性条件。我们把重点放在特定种类的曲率条件，我们称之为非强制性，这些条件的非负曲率和消失的数量曲率并不意味着平坦。我们证明，在大于4的维度，如果一个里奇流不变条件是弱于“爱因斯坦与非负数量曲率“，那么这个条件必须是“非负数量曲率“。作为推论，我们得到了紧Einstein对称空间如S^2 xS^2或CP ^2不能（严格）满足比“非负数量曲率“更强的里奇流不变曲率条件.我们还研究了所有具有非负数量曲率的共形平坦流形所满足的条件。
Splitting theorems for Finsler manifolds of nonnegative Ricci curvature
研究了包含直线的非负加权里奇曲率的Finsler流形的结构，推广了经典的Cheeger-Gromoll-Lichnerowicz分裂定理.这样一个空间一般容许一个保纯的、保测度的分裂。对于一类特殊的Berwald空间，当Busemann函数的梯度向量场生成一个单参数等距族时，我们可以进行等距分裂.对Berwald空间也给出了一个贝蒂数估计.
Sharpening a gap theorem: nonnegative Ricci and small curvature concentration
本研究强化了非负Ricci曲率流形的间隙定理，证明曲率集中只需与渐近体积比线性相关，通过构造快于$1/t$衰减的长时间Ricci流解得到间隙的显式界。
Four-dimensional gradient Ricci solitons with (half) nonnegative isotropic curvature
本文作为此前研究的续作，针对具有非负或半非负迷向曲率的四维梯度稳态、膨胀Ricci孤立子得到分类相关结果，并改进了梯度收缩Ricci孤立子的已有结论，给出复二维Kaehler-Ricci收缩子的刻画。
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。
Fundamental groups of open manifolds with nonnegative Ricci curvature and universal cover Euclidean volume growth
本研究针对非负Ricci曲率开流形的基本群性质问题，证明了万有覆叠具欧氏体积增长时基本群有限生成且实质上阿贝尔，还将有限生成结论推广到更一般条件，肯定了Pan-Rong猜想。
A splitting theorem for manifolds with spectral nonnegative Ricci curvature and mean-convex boundary
该研究证明了具有非负谱瑞奇曲率和均凸边界的非紧流形要么等距于Σ×R_{≥0}，要么没有内部端点。
On manifolds with nonnegative Ricci curvature and the infimum of volume growth order $<2$
本文针对具有非负Ricci曲率且体积增长阶下确界小于2的开流形，证明了两个刚性定理：万有覆盖具有欧氏体积增长等价于流形平坦且具有n-1维soul；存在非常值线性增长调和函数等价于流形等距同构于实数线与紧流形的度量积。
A Polyharmonic Liouville Hierarchy on Complete Manifolds of Nonnegative Ricci Curvature
本文在非负里奇曲率流形上建立了多项调和李乌利维定理，通过L²估计证明亚线性增长的多项调和函数必为常数，拓展了欧几里得空间的分类结果。
Integrals and Rigidity on Manifolds with Nonnegative Ricci Curvature
本文研究了非负里奇曲率流形上的积分不等式及刚性问题，推导出标量曲率的渐进行度不变积分公式，并用此公式证明了Hamilton偏心猜想。
A consequence of the Cheeger-Gromoll splitting theorem states that for any open manifold $(M,x)$ of nonnegative Ricci curvature, if all the minimal geodesic loops at $x$ that represent elements of $π_1(M,x)$ are contained in a bounded ball, then $π_1(M,x)$ is virtually abelian. We generalize the above result: if these minimal representing geodesic loops of $π_1(M,x)$ escape from any bounded metric balls at a sublinear rate with respect to their lengths, then $π_1(M,x)$ is virtually abelian.
We prove a lower bound for the first Steklov eigenvalue of embedded minimal hypersurfaces with free boundary in a compact $n$-dimensional manifold which has nonnegative Ricci curvature and strictly convex boundary. When $n=3$, this implies apriori area and curvature estimates for these minimal surfaces in terms of the geometry of the ambient manifold and the topology of the minimal surface. An important consequence of the estimate is a smooth compactness result for embedded minimal surfaces with free boundary when the topological type of these minimal surfaces is fixed.
In this paper, we classify graphs with nonnegative Lin-Lu-Yau-Ollivier Ricci curvature, maximum degree at most 3 and diameter at least 6.
Collapsing sequences of solutions to the Ricci flow on 3-manifolds with almost nonnegative curvature
本文研究截面曲率几乎非负且直径趋于无穷大的里奇流的三维解序列。这样的序列可能是由关于IIb型奇点的膨胀极限引起的。特别是，我们研究的情况下，序列崩溃，这可能会发生在无限时间奇点膨胀。在这种情况下，我们分类可能的Gromov-Hausdorff极限和构造2维虚拟极限。利用局部覆盖的极限定理构造虚极限。然后，我们表明，所产生的虚拟限制，从适当的膨胀的IIb型奇点总是汉密尔顿的雪茄孤子解决方案。
Nonnegative Ricci curvature, small linear diameter growth and finite generation of fundamental groups
1968年，Milnor证明了具有非负里奇曲率的完备非紧流形有一个非负生成的基本群.作者应用Abresch和Gromoll（1990）的过剩定理证明了两个定理。第一个状态，如果这样的流形有小的线性直径增长，那么它的基本群是连续生成的。第二个国家，如果这样的流形有一个无限生成的基本群，然后它有一个切锥在无穷远这是不极性。这两个定理的一个推论是，如果这样的流形具有线性体积增长，那么它的基本群是线性生成的。
We prove that $n$-dimensional ($n\geqslant3$) complete and non-compact metric measure spaces with non-negative weighted Ricci curvature in which some Caffarelli-Kohn-Nirenberg type inequality holds are close to the model metric measure $n$-space (i.e., the Euclidean metric $n$-space).
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 $4k+3\geq 7$.
Liouville theorem for biharmonic functions on manifolds of nonnegative Ricci curvature
本文证明了在非负瑞奇曲率流形中，具有次二次增长的双曲函数必须为调和函数，而具有次线性增长的则为常数，扩展了Yau的李普希兹定理。
Nonnegative Ricci curvature, metric cones, and virtual abelianness
本文研究非负 Ricci 曲率开流形的基本群，在逃逸率非 1/2 且万有覆盖渐近为度量锥时证明其虚拟阿贝尔性，并在欧氏体积增长常数至少为 L 时给出指数界 C(n,L)。
Metrics of positive Ricci curvature on bundles
我们构造了正里奇曲率流形的新例子，拓扑上，几乎非负里奇曲率紧致流形上的向量丛。特别地，我们证明了：如果E是紧致流形上具有非负里奇曲率的向量丛的全空间，则E与R^p的乘积对所有大p都存在正里奇曲率的完备度量.
Suppose $(M,g)$ is a Riemannian manifold having dimension $n$, nonnegative Ricci curvature, maximal volume growth and unique tangent cone at infinity. In this case, the tangent cone at infinity $C(X)$ is an Euclidean cone over the cross-section $X$. Denote by $α=\lim_{r\rightarrow\infty}\frac{\mathrm{Vol}(B_{r}(p))}{r^{n}}$ the asymptotic volume ratio. Let $h_{k}=h_{k}(M)$ be the dimension of the space of harmonic functions with polynomial growth of growth order at most $k$. In this paper, we prove a upper bound of $h_{k}$ in terms of the counting function of eigenvalues of $X$. As a corollary…
On Loops Representing Elements of the Fundamental Group of a Complete Manifold with Nonnegative Ricci Curvature
本文研究具有非负里奇曲率的完备非紧流形.粗略地说，如果给定任意不可压缩的闭曲线C和任意紧集K，存在包含在M \K中的闭曲线与C同伦，则称M具有无穷大循环性质.本文的主要定理如下。定理I：若M具有正的里奇曲率，则它具有无穷大环性质.定理二：若M具有非负里奇曲率，则它要么具有无穷大环性质，要么等距于紧致全测地子流形上的平坦法丛，且其双覆盖是分裂的.定理三：设M是完备的黎曼流形，M具有从点p开始的沿某条射线到无穷远的环，D是包含点p的具有光滑边界的预紧区域，S是射线上包含点q的边界的任意连通分支.则由包含映射导出的从S的以q为基的基本群到Cl（D）的以p为基的基本群的映射为上。
Compact Mean Convex Hypersurfaces and the Fundamental Group of Manifolds with Nonnegative Ricci Curvature
证明了在非负里奇曲率的非紧连通完备黎曼n-流形N中存在嵌入的紧致无边界严格正中曲率超曲面S，这意味着由包含所诱导的S和N的基本群之间的同态是满射的，只要N-S有两个连通分支，其中一个分支相对于S具有非紧闭包和平凡同伦.证明的思想是将N看作一个合适的洛伦兹流形中的类空超曲面，然后应用甘农和李关于时空拓扑的某些经典结果的最新版本。作为应用，我们证明了如果N是渐近平坦的，那么它只有一端，并且当n大于3时，N是单连通的。
On the topology of manifolds with nonnegative Ricci curvature and linear volume growth
本文研究非负 Ricci 曲率与线性体积增长的开流形，证明其基本群含有限指数子群 $\mathbb{Z}^k$，且正 Ricci 曲率下基本群有限。
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