Mingliang Cai, Greg Galloway
The following version of a conjecture of Fischer-Colbrie and Schoen is proved: If M is a complete Riemannian 3-manifold with nonnegative scalar curvature which contains a two-sided torus S which is of least area in its isotopy class then M is flat. This follows from a local version derived in the paper.
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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.
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.
In this paper, we are going to show some rigidity results for complete open Riemannian manifolds with nonnegative scalar curvature. Without using the famous Cheeger-Gromoll splitting theorem we give a new proof to a rigidity result for complete manifolds with nonnegative scalar curvature admitting a proper smooth map to $T^{n-1}\times \mathbf R$ with nonzero degree. Here we introduce a trick to obtain the compactness of limit hypersurface from locally graphical convergence. Based on the same idea we also establish an optimal $2$-systole inequality for several classes of complete Riemannian man…
We show that one-dimensional circle is the only case for closed smooth metric measure spaces with nonnegative Bakry-Émery Ricci curvature whose spectrum of the weighted Laplacian has an optimal positive upper bound. This result extends the work of Hang-Wang in the manifold case (Int. Math. Res. Not. 18 (2007), Art. ID rnm064, 9pp).
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。
Let M be a closed minimal hypersurface in 5-dimensional Euclidean sphere with constant nonnegative scalar curvature. We prove that, if the sum of the cubes of all principal curvatures and the number of distinct principal curvatures are constant, then M is isoparametric. Moreover, We give all possible values for squared length of the second fundamental form of M. This result provides another piece of supporting evidence to the Chern conjecture.
Let $(M^n,g)$, $n \ge 4$, be a compact simply-connected Riemannian manifold with nonnegative isotropic curvature. Given $0<l\le L$, we prove that there exists $\eps = \eps (l,L,n)$ satisfying the following: If the scalar curvature $s$ of $g$ satisfies $$ l \le s \le L $$ and the Einstein tensor satisfies $$ | Ric - \frac {s}{n}g | \le \eps$$ then $M$ is diffeomorphic to a symmetric space of compact type. This is a smooth analogue of the result of S. Brendle that a compact Einstein manifold with nonnegative isotropic curvature is isometric to a locally symmetric space.
Topological Rigidity of Contractible 3-Manifolds and Handlebody Interiors under Nonnegative Scalar Curvature
研究非负标量曲率下可缩三维流形及柄体内部拓扑刚性，通过证明可缩三维流形与\(\mathbb{R}^3\)微分同胚等，回答了王和格罗莫夫的两个开放性问题。
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。
Scalar-mean rigidity beyond warped product spaces
本文提出了一类具有边界的黎曼流形的标量-均值极值和刚性定理，扩展了传统扭曲积空间的范围，并引入了新的刚性流形家族。
Topological and geometric rigidity of nonnegatively curved submanifolds
研究非负曲率空间形式中满足特定条件的紧致子流形，通过特定方法探究其拓扑与几何，证明条件对其拓扑或几何有强约束并给出精确性例子。
Given a closed Riemannian manifold (M, g), there is a partition Σ_i of its tangent bundle TM called the focal decomposition. The sets Σ_i are closely associated to focusing of geodesics of (M, g), i.e. to the situation where there are exactly i geodesic arcs of the same length joining points p and q in M. In this note, we study the topological structure of the focal decomposition of a closed Riemannian manifold and its relation with the metric structure of the manifold. Our main result is that the flat n-tori are focally rigid, in the sense that if two flat tori are focally equivalent, then th…
Nonnegative Ricci curvature, splitting at infinity, and first Betti number rigidity
本文研究非负里奇曲率开流形的刚性，证明渐近锥包含欧氏空间时第一贝蒂数受维数差控制且等号情形流形平坦，并给出轨道增长阶的几何刚性及首个增长阶可变的例子。
Singularity removal rigidity theorems for minimal hypersurfaces in manifolds with nonnegative scalar curvature
本文证明了非负标量曲率流形中极小超曲面的奇点去除刚性定理，并利用该结果直接证明了渐近扁$8$-流形的正质量定理。
In this paper, we prove some rigidity theorems for shrinking gradient Ricci solitons with nonnegative sectional curvature.
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。
By works of Schoen-Yau and Gromov-Lawson any Riemannian manifold with nonnegative scalar curvature and diffeomorphic to a torus is isometric to a flat torus. Gromov conjectured subconvergence of tori with respect to a weak Sobolev type metric when the scalar curvature goes to $0$. We prove flat and intrinsic flat subconvergence to a flat torus for noncollapsing sequences of $3$-dimensional tori $M_j$ that can be realized as graphs of certain functions defined over flat tori satisfying a uniform upper diameter bound and scalar curvature bounds of the form $R_{g_{M_j}} \geq -1/j$. We also show t…
Rigidity of the timelike marked length spectrum and length-twist coordinates of singular de-Sitter tori
本文研究了具有一个奇点的德西特环形的闭合类时测地线，证明其在自由同调类中唯一。引入了此类环形的时样标记长度谱概念，并证明其在两个交数为一的同调类长度上的刚性。还构造了具有一个奇点的德西特环形变形空间上的长度-扭坐标。
Rigidity of singular de-Sitter tori with respect to their lightlike bi-foliation
本文研究了具有锥形奇点的洛伦兹面的刚性，证明了德西特环面由其光状双分层的拓扑等价类确定。
We prove a splitting theorem for complete gradient Ricci soliton with nonnegative curvature and establish a rigidity theorem for codimension one complete shrinking gradient Ricci soliton in $\mathbb R^{n+1}$ with nonnegative Ricci curvature.
We show that sufficiently irreducible Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions.
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 $\mathbb{R}^n$。
Scalar curvature rigidity of degenerate warped product spaces
该研究证明了一类两端可能退化的扭积空间的标量曲率极值性与刚性，涵盖多种叶流形情形，正面解答了Gromov在所有维数下的相关问题。
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.
Conformal structures and rigidity of complete stable minimal hypersurfaces
本文研究不同曲率条件下完备稳定极小超曲面的共形结构与刚性，证明非负数量曲率流形中不存在共形等价于非正数量曲率有界区域的非紧稳定极小超曲面，并给出紧致情形及扭曲积流形中的稳定性与刚性结果。
In this paper we prove a rigidity result for the equality case of the Penrose inequality on $3$-dimensional asymptotically flat manifolds with nonnegative scalar curvature and corners. Our result also has deep connections with the equality cases of Theorem 1 in \cite{Miao2} and Theorem 1.1 in \cite{LM}.
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。
It is well-know that Hawking mass is nonnegative for a stable constant mean curvature ($CMC$) sphere in three manifold of nonnegative scalar curvature. R. Bartnik proposed the rigidity problem of Hawking mass of stable $CMC$ spheres. In this paper, we show partial rigidity results of Hawking mass for stable $CMC$ spheres in asymptotic flat ($AF$) manifolds with nonnegative scalar curvature. If the Hawking mass of a nearly round stable $CMC$ surface vanishes, then the surface must be standard sphere in $\mathbb{R}^3 $ and the interior of the surface is flat. The similar results also hold for as…
Mass-Capacity Rigidity for Asymptotically Flat Manifolds with Arbitrary Ends
本文刻画了任意端渐近平坦流形质量-容量不等式等号成立的条件，证明等号迫使流形等距于欧氏空间去掉低维紧集，关键利用共形变换与Bakry-Émery曲率。
Rigidity of CMC hypersurfaces in 5-and 6-manifolds
本文证明在5维和6维流形中，非负中间里奇曲率等曲率条件下，稳定极小超曲面及非零平均曲率常平均曲率超曲面的刚性，推广了Chodosh-Li-Stryker的结果。
We consider complete noncompact Riemannian manifolds with quadratically decaying lower Ricci curvature bounds and minimal volume growth. We first prove a rigidity result showing that ends with strongly minimal volume growth are isometric to warped product manifolds. Next we consider the almost rigid case in which manifolds with nonnegative and quadratically decaying lower Ricci curvature bounds have minimal volume growth. Compact regions in such manifolds are shown to be asymptotically close to warped products in the Gromov-Hausdorff topology. Manifolds with nonnegative Ricci curvature and lin…
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.
Scalar curvature comparison and rigidity of $3$-dimensional weakly convex domains
本文证明了具有平均凸边界的紧致黎曼3-流形在标量曲率非负且满足缩放平均曲率比较时是平坦的，并将结果推广至带角流形，其证明利用了毛细极小曲面和叶状结构。
We study nonnegatively curved metrics on S^2xR^4. First, we prove rigidity theorems for connection metrics; for example, the holonomy group of the normal bundle of the soul must lie in a maximal torus of SO(4). Next, we prove that Wilking's almost-positively curved metric on S2xS3 extends to a nonnegatively curved metric on S^2xR^4 (so that Wilking's space becomes the distance sphere of radius 1 about the soul). We describe in detail the geometry of this extended metric.
We prove the following comparison theorem for metrics with nonnegative scalar curvature, also known as the dihedral rigidity conjecture by Gromov: for $n\le 7$, if an $n$-dimensional prism has nonnegative scalar curvature and weakly mean convex faces, then its dihedral angle cannot be everywhere not larger than its Euclidean model, unless it is isometric to an Euclidean prism. The proof relies on constructing certain free boundary minimal hypersurface in a Riemannian polyhedron, and extending a dimension descent idea of Schoen-Yau. Our result is a localization of the positive mass theorem.
New Curvature Operator of the Second Kind Conditions for Rigidity
本文通过第二类曲率算子建立Tachibana型刚性定理，证明维数≥5的完备Einstein流形在$C_n$-非负条件下具有常非负截面曲率，并推广至平移算子与对称空间分类。
We show that complete conformally flat manifolds of dimension n>2 with nonnegative Ricci curvature enjoy nice rigidity properties: they are either flat, or locally isometric to a product of a sphere and a line, or are globally conformally equivalent to flat space or to a spherical spaceform. This extends previous works by Q.-M. Cheng, M.H. Noronha, B.-L. Chen and X.-P. Zhu, and S. Zhu.
We prove a new rigidity result for an open manifold M with nonnegative sectional curvature whose soul S is odd-dimensional. Specifically, there exists a geodesic in S and a parallel vertical plane field along it with constant vertical curvature and vanishing normal curvature. Under the added assumption that the Sharafutdinov fibers are rotationally symmetric, this implies that for small r, the distance sphere of radius r about S contains an immersed flat cylinder, and thus could not have positive curvature.
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.
In this article, we investigate deformation problems of $Q$-curvature on closed Riemannian manifolds. One of the most crucial notions we use is the $Q$-singular space, which was introduced by Chang-Gursky-Yang during 1990's. Inspired by the early work of Fischer-Marsden, we derived several results about geometry related to $Q$-curvature. It includes classifications for nonnegative Einstein $Q$-singular spaces, linearized stability of non-$Q$-singular spaces and a local rigidity result for flat manifolds with nonnegative $Q$-curvature. As for global results, we showed that any smooth function c…
Scalar Curvature Rigidity for Products of convex Hypersurfaces and even dimensional Manifolds
研究欧几里得空间中严格凸超曲面与特定非负曲率空间乘积的标量曲率刚性，基于弗雷德霍姆族指标定理给出证明，恢复了相关结果。
We derive a curvature-variation formula for a path of left-invariant metrics on a compact Lie group, beginning at a bi-invariant metric. We prove rigidity theorems for paths which remain nonnegatively curved, and we make progress towards a classification of the left-invariant metrics with nonnegative curvature on SO(4).
Quasilinear Liouville equation on manifolds with nonnegative Ricci curvature
研究非负Ricci曲率完备非紧Riemann流形上n-Laplace拟线性Liouville方程的刚性与分类，证明在尖锐对数下界条件下流形等距于欧氏空间且解为标准气泡，并构造非平坦流形上的解表明假设的尖锐性。
Sharp rigidity for quasilinear Liouville equation on manifolds with nonnegative Ricci curvature
研究具有非负Ricci曲率流形上的拟线性Liouville方程，通过解的最优对数下界得出流形与欧氏空间等距且解为标准泡解，证明中关键是对数下界与总体积上界联系，还提出并证明了关于解与几何相互作用的猜想。
On some rigidity theorems of Q-curvature
本文研究了闭定向黎曼流形在特定积分条件下的Q-曲率刚性，证明了局部共形平坦及非负曲率条件下的等距分类定理，并探讨了超曲面刚性与固定共形类中常曲率度量的唯一性。
Let $F^{n+p}(c)$ be an $(n+p)$-dimensional simply connected space form with nonnegative constant curvature $c$. We prove that if $M^n(n\geq4)$ is a compact submanifold in $F^{n+p}(c)$, and if $Ric_M>(n-2)(c+H^2),$ where $H$ is the mean curvature of $M$, then $M$ is homeomorphic to a sphere. We also show that the pinching condition above is sharp. Moreover, we obtain a new differentiable sphere theorem for submanifolds with positive Ricci curvature.
The rigidity of sharp spectral gap in nonnegatively curved spaces
该研究将非负Ricci曲率下的第一尖锐谱隙刚性推广到紧RCD(0,N)空间，证明谱隙取极值等价于空间为一维常密度，所用的Sobolev理论与奇异一维局部化结合的新技术有独立价值，并导出几乎刚性结果。
Some rigidity theorems for spectral curvature bounds
本文探讨了谱曲率界在几何中的意义，扩展了标量曲率几何中的经典刚性结果到谱设置，通过系统应用扭曲μ-气泡方法，对非负谱标量曲率3流形中的稳定加权极小超曲面进行分类，并建立了谱Ricci和谱标量曲率的带宽估计。
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。
Rigidity of Schouten Tensor under Conformal Deformation
本文研究了共形形变下 Schouten 张量有下界的度量刚性，利用该张量非负与 Ricci pinching 条件的等价性，为 Cheng 关于局部共形平坦流形的紧致性定理提供了简单证明。
Level sets of harmonic functions on three-dimensional manifolds with nonnegative scalar curvature
本文研究非负数量曲率三维流形上调和函数水平集的单调量，建立刚性结果，并发现额外数量平坦度量。
In this paper we prove an area comparison result for certain totally geodesic surfaces in 3-manifolds with a lower bound on the scalar curvature. This result is a variant of a comparison theorem of Heintze-Karcher for minimal hypersurfaces in manifolds of nonnegative Ricci curvature. Our assumptions on the ambient manifold are weaker but the assumptions on the surface are considerably more restrictive. We then use our comparison theorem to provide a unified proof of various splitting theorems for 3-manifolds with lower bounds on the scalar curvature.
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…
Scalar curvature rigidity for products of spheres and tori
本文证明了球面与环面乘积上的 Llarull 型标量曲率刚性，并利用稳定加权切片与谱 Dirac 算子方法推广 Gromov 环面不等式，得到带状区域的尖锐宽度界。
In this paper we prove classification results for solutions to subcritical and critical semilinear elliptic equations with a nonnegative potential on noncompact manifolds with nonnegative Ricci curvature. We show in the subcritical case that all nonnegative solutions vanish identically. Moreover, under some natural assumptions, in the critical case we prove a strong rigidity result, namely we classify all nontrivial solutions showing that they exist only if the potential is constant and the manifold is isometric to the Euclidean space.
A real Bott manifold is the total space of iterated RP^1 bundles starting with a point, where each RP^1 bundle is projectivization of a Whitney sum of two real line bundles. We prove that two real Bott manifolds are diffeomorphic if their cohomology rings with Z/2 coefficients are isomorphic. A real Bott manifold is a real toric manifold and admits a flat riemannian metric invariant under the natural action of an elementary abelian 2-group. We also prove that the converse is true, namely a real toric manifold which admits a flat riemannian metric invariant under the action of an elementary abe…
Rigidity of positively curved Steady gradient Ricci solitons on orbifolds
本文研究轨道流形上正曲率稳态梯度Ricci孤立子的刚性，证明其数量曲率非负，并在正曲率算子或正截面曲率条件下，此类孤立子必须是Bryant孤立子的有限商。
Divergence Identity for the scalar curvature and Rigidity of Codazzi Tensors
本文推导了标量曲率的发散恒等式，并应用于柯西-阿扎尔对称张量的刚性问题，证明了在非负标量曲率条件下，具有特定迹不变量的张量必须具有常数特征值。
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.
In this paper we generalize the main result of [4] for manifolds that are not necessarily Einstein. In fact, we obtain an upper bound for the volume of a locally volume-minimizing closed hypersurface $Σ$ of a Riemannian 5-manifold $M$ with scalar curvature bounded from below by a positive constant in terms of the total traceless Ricci curvature of $Σ$. Furthermore, if $Σ$ saturates the respective upper bound and $M$ has nonnegative Ricci curvature, then $Σ$ is isometric to $\mathbb{S}^4$ up to scaling and $M$ splits in a neighborhood of $Σ$. Also, we obtain a rigidity result for the Riemannian…
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