Hongbing Qiu, Linlin Sun
In this paper, we firstly establish a new volume growth estimate for spacelike entire graphs in the pseudo-Euclidean space $\mathbb{R}^{m+n}_n$. Then by using this volume growth estimate and the Co-Area formula, we prove various rigidity results for spacelike entire self-shrinking graphs.
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I classify spacelike self-similar shrinking solutions of the mean curvature flow in pseudo-euclidean space in arbitrary codimension, if the mean curvature vector is not a null vector and the principal normal vector is parallel in the normal bundle. Moreover, I exclude the existence of such self-shrinkers in several cases. The classification is analogous to the existing classifications in the euclidean case of Huisken and Smoczyk.
We prove rigidity theorems for shrinking gradient Ricci solitons supporting the Heisenberg-Pauli-Weyl uncertainty principle with the sharp constant in $\mathbb{R}^n$. In addtion, we partially give analogous rigidity results of the Caffarelli-Kohn-Nirenberg inequalities on shrinking Ricci solitons.
In this paper, we prove some rigidity results for both shrinking and expanding Ricci solitons. First, we prove that compact shrinking Ricci solitons are Einstein if we control the maximum value of the potential function. Then, we prove some rigidity results for non-compact gradient expanding and shrinking Ricci solitons with pinched Ricci and scalar curvatures, assuming an asymptotic condition on the scalar curvature at infinity.
We show that every entire self-shrinking solution on $\mathbb{C}^1$ to the Kähler-Ricci flow must be generated from a quadratic potential.
In this paper, we investigate the parametric version and non-parametric version of rigidity theorem of spacelike translating solitons in pseudo-Euclidean space $\mathbb{R}^{m+n}_{n}$. Firstly, we classify $m$-dimensional complete spacelike translating solitons in $\mathbb{R}^{m+n}_{n}$ by affine technique and classical gradient estimates, and prove the only complete spacelike translating solitons in $\mathbb{R}^{m+n}_{n}$ are the spacelike $m$-planes. This result provides another proof of a nonexistence theorem for complete spacelike translating solitons in \cite{C-Q}, and a simple proof of ri…
We prove that all entire smooth strictly convex self-shrinking solutions on $\mathbb{R}^n$ to the Hessian quotient flows must be quadratic. This generalizes the rigidity theorem for entire self-shrinking solutions to the Lagrangian mean curvature flow in pseudo-Euclidean space due to Ding-Xin \cite{DX}. Moreover, we show that our argument works for a larger class of equations. In particular, we obtain rigidity results for entire self-shrinking solutions on $\mathbb{C}^n$ to the Kähler-Ricci flow under certain conditions.
In this paper we present a rigidity theorem for locally isometric hypersurfaces with a curvature restriction in de Sitter space. This is an analogue to the case for Riemannian space forms given by Guan and Shen in [5].
New rigidity results for complete non-compact spacelike submanifolds of arbitrary codimension in plane fronted waves are obtained. Under appropriate assumptions, we prove that a complete spacelike submanifold in these spacetimes is contained in a characteristic lightlike hypersurface. Moreover, for a complete codimension two extremal submanifold in a plane fronted wave we show sufficient conditions to guarantee that it is a (totally geodesic) wavefront.
We prove that an $n$-dimensional, $n\geq4$, compact gradient shrinking Ricci soliton satisfying a $L^{\frac n2}$-pinching condition is isometric to a quotient of the round $\mathbb{S}^n$, which improves the rigidity theorem given by G. Catino (arXiv:1509.07416vl).
Compactness and rigidity of self-shrinking surfaces
本文利用测度论技术与刚性结果，证明了低熵自收缩曲面的紧致性定理，进而确立熵极小化子的存在性并改进刚性结论。
We classify four-dimensional shrinking Ricci solitons satisfying $Sec \geq \frac{1}{24} R$, where $Sec$ and $R$ denote the sectional and the scalar curvature, respectively. They are isometric to either $\mathbb{R}^{4}$ (and quotients), $\mathbb{S}^{4}$, $\mathbb{RP}^{4}$ or $\mathbb{CP}^{2}$ with their standard metrics.
We prove that every entire self-shrinking solution on $\mathbb{C}^n$ to the Kähler-Ricci flow with strictly real convex potential must be quadratic. The very same argument also gives a pointwise proof for the rigidity of entire self-shrinking solutions to Lagrangian mean curvature flow in pseudo-Euclidean space obtained by Q. Ding and Y.L. Xin. Furthermore, we show that our argument works for a larger class of equations.
We prove that a gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is rigid. For the $4$-dimensional case, we show that any gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is either Einstein, or a finite quotient of the Gaussian shrinking soliton $\mathbb{R}^4$, $\mathbb{R}^2\times\mathbb{S}^2$ or the round cylinder $\mathbb{R}\times\mathbb{S}^3$. Under the condition of fourth order divergence-free Weyl tensor, we have the same results.
Rigidity of Five-Dimensional shrinking gradient Ricci solitons
本文证明了曲率有界且标量曲率为1的五维完备收缩梯度Ricci孤立子必为 $\mathbb{R}^3\times \mathbb{S}^2$ 的有限商，揭示了该类几何对象的刚性结构。
Rigidity Results for Spacelike Self-Shrinkers via Different Maximum Principles
本文在伪欧几里得空间$\mathbb{R}^{n+p}_p$中，利用不同版本的Omori--Yau型最大值原理，在平均曲率向量或第二基本形式有界的条件下，证明类空自收缩子必为类空超平面，为其分类提供了刚性结果。
Rigidity of spacelike hypersurface with constant curvature and intersection angle condition
研究类空超曲面在Minkowski空间中的刚性性质，证明当k阶均曲率常数且边界与超平面交角恒定时，超曲面必为双曲面的一部分或完全包含于超平面中。
Let $(M, g, f)$ be a $4$-dimensional complete noncompact gradient shrinking Ricci soliton with the equation $Ric+\nabla^2f=λg$, where $λ$ is a positive real number. We prove that if $M$ has constant scalar curvature $S=2λ$, it must be a quotient of $\mathbb{S}^2\times \mathbb{R}^2$. Together with the known results, this implies that a $4$-dimensional complete gradient shrinking Ricci soliton has constant scalar curvature if and only if it is rigid, that is, it is either Einstein, or a finite quotient of Gaussian shrinking soliton $\Bbb{R}^4$, $\Bbb{S}^{2}\times\Bbb{R}^{2}$ or $\Bbb{S}^{3}\time…
Our purpose in this paper is to apply some maximum principles in order to study the rigidity of complete spacelike hypersurfaces immersed in a spatially weighted generalized Robertson-Walker (GRW) spacetime, which is supposed to obey the so called strong null convergence condition. Under natural constraints on the weight function and on the f-mean curvature, we establish sufficient conditions to guarantee that such a hypersurface must be a slice of the ambient space. In this setting, we also obtain new Calabi-Bernstein type results concerning entire graphs in a spatially weighted GRW spacetime…
Some rigidity results on shrinking gradient Ricci soliton
本文研究完备收缩梯度Ricci孤立子的刚性，利用极大值原理证明常数量曲率$R=1$时等距于$\mathbb{R}^2\times\mathbb{S}^2$的有限商，推广了已有结果并给出Cheng-Zhou定理的新证明。
Rigidity and nonexistence of complete spacelike hypersurfaces in the steady state space
本文研究稳态空间中完备类空超曲面，在适当高阶平均曲率约束下证明其必为类空超平面，并给出不存在性结果，方法基于广义最大值原理。
Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature
本文研究五维完备非紧梯度收缩里奇孤立子，证明当数量曲率为常数且等于 3λ 时，该流形必为 R²×S³ 的有限商，从而给出该类孤立子的刚性分类。
Rigidity of eigenvalues of shrinking Ricci solitons
本文研究收缩Ricci孤立子的特征值刚性，证明了第n个特征值接近下界时孤立子为平凡高斯孤立子，在非坍塌条件下得到第(n-1)、(n-2)个特征值的类似结果，并给出一般k的第k个特征值的几乎刚性结论，部分结果拓展了紧流形上的相关定理至非紧孤立子情形。
We show that every complete entire self-shrinking solution on complex Euclidean space to the Kahler-Ricci flow must be generated from a quadratic potential.
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.
Transverse Rigidity of Shrinking Sasaki-Ricci Solitons
本文研究了Sasaki-Ricci溶iton作为Sasaki-Ricci流奇点模型的性质，建立了基本方程并证明了标量曲率正性，提出两个横向刚性判据，指出常标量曲率低维溶iton必为Sasaki-Einstein，且调和Weyl张量溶iton为球面有限商。
Rigidity of stable spacelike capillary hypersurfaces in de Sitter and Minkowski spaces
该研究证明了德西特与闵可夫斯基空间中，当支撑为非负内蕴曲率的类时全脐超曲面时，保体积稳定的紧致类时毛细管超曲面必为全脐，还推导了统一闵可夫斯基型公式并构造了测试族。
In this paper, we prove some rigidity theorems for shrinking gradient Ricci solitons with nonnegative sectional curvature.
We prove rigidity of any properly immersed noncompact Lagrangian shrinker with single valued Lagrangian angle for Lagrangian mean curvature flows. Our pointwise approach also provides an ele- mentary proof to the known rigidity results for graphical and almost graphical shrinkers of mean curvature flows.
We show that (a) any entire graphic self-shrinking solution to the Lagrangian mean curvature flow in ${\mathbb C}^{m}$ with the Euclidean metric is flat; (b) any space-like entire graphic self-shrinking solution to the Lagrangian mean curvature flow in ${\mathbb C}^{m}$ with the pseudo-Euclidean metric is flat if the Hessian of the potential is bounded below quadratically; and (c) the Hermitian counterpart of (b) for the Kähler Ricci flow.
We study the rigidity results for self-shrinkers in Euclidean space by restriction of the image under the Gauss map. The geometric properties of the target manifolds carry into effect. In the self-shrinking hypersurface situation Theorem 3.1 and Theorem 3.2 not only improve the previous results, but also are optimal. In higher codimensional case, using geometric properties of the Grassmanian manifolds (the target manifolds of the Gauss map) we give a rigidity theorem for self-shrinking graphs.
We show that $S^2\times S^2$ is isolated as a shrinking Ricci soliton in the space of metrics, up to scaling and diffeomorphism. We also prove the same rigidity for $S^2\times N$, where $N$ belongs to a certain class of closed Einstein manifolds. These results are the Ricci flow analogues of our results for Clifford-type shrinking solitons for the mean curvature flow.
Geometric inequalities and rigidity of gradient shrinking Ricci solitons
该研究证明完备梯度收缩 Ricci 孤立子上六类几何不等式与热核估计等价存在，并给出紧致情形下的积分间隙刚性结果。
We show that if two gradient Ricci solitons are asymptotic along some end of each to the same regular cone, then the soliton metrics must be isometric on some neighborhoods of infinity of these ends. Our theorem imposes no restrictions on the behavior of the metrics off of the ends in question and in particular does not require their geodesic completeness. As an application, we prove that the only complete connected gradient shrinking Ricci soliton asymptotic to a rotationally symmetric cone is the Gaussian soliton on R^n.
Rigidity of shrinking gradient ricci soliton with constant scalar curvature
该研究针对具有常数量曲率的完备收缩梯度Ricci孤立子，在两个假设条件下证明其等距于$\mathbb{R}^2 \times \mathbb{S}^{n-2}$，且所用条件弱于已有径向平坦条件。
New general results of non-existence and rigidity of spacelike submanifolds immersed in a spacetime, whose mean curvature is a time-oriented causal vector field, are given. These results hold for a wide class of spacetimes which includes globally hyperbolic, stationary, conformally stationary and pp-wave spacetimes, among others. Moreover, applications to the Cauchy problem in General Relativity, are presented. Finally, in the case of hypersurfaces, we also obtain significant consequences in Geometrical Analysis, solving new Calabi-Bernstein and Dirichlet problems on a Riemannian manifold.
Compact spacelike biconservative hypersurfaces in de Sitter space
本文研究de Sitter空间中常数量曲率紧致类空双守恒超曲面在几何约束下的刚性性质，拓展了伪黎曼背景下该类超曲面的理解。
High codimension mean curvature flow of spacelike-convex submanifolds with one spacelike codimension
研究伪欧氏空间中高码维类空子流形的平均曲率流，证明初始紧致类空凸子流形在流下曲率收缩和非坍缩量保持，并证明Huisken和Gage-Hamilton定理的类比结果。
There are several well-known characterizations of the sphere as a regular surface in the Euclidean space. By means of a purely synthetic technique, we get a rigidity result for the sphere without any curvature conditions, nor completeness or compactness. As well as a dual result for the hyperbolic plane, the spacelike sphere in the Minkowski space.
In this note we show that compact self shrinkers in $\mathbb{R}^3$ are "topologically standard" in that any genus $g$ compact self shrinker is ambiently isotopic to the standard genus $g$ embedded surface in $\mathbb{R}^3$. As a consequence self shrinking tori are unknotted.
We show that the product of two round shrinking spheres is an isolated self-shrinker in any codimension, modulo rotations. Moreover we prove explicit Łojasiewicz inequalities near such products. Łojasiewicz inequalities were previously used by Schulze to prove uniqueness of tangent mean curvature flows at compact shrinkers; our results provide an explicit rate of convergence to products of two spheres.
Rigidity of Gradient Shrinking Ricci Solitons with a Vanishing Bach-like Tensor and Related Variational Formulas
研究四维梯度收缩黎曼流形在 Bach-like 张量消失时的刚性性质，并探讨其与二次曲率函数的变分关系。
By the integral method we prove that any space-like entire graphic self-shrinking solution to Lagrangian mean curvature flow in $\R^{2n}_{n}$ with the indefinite metric $\sum_i dx_idy_i$ is flat. This result improves the previous ones in \cite{HW} and \cite{CCY} by removing the additional assumption in their results. In a similar manner, we reprove its Euclidean counterpart which is established in \cite{CCY}.
In this paper we investigate the rigidity of ancient solutions of the mean curvature flow with arbitrary codimension in space forms. We first prove that under certain sharp asymptotic pointwise curvature pinching condition the ancient solution in a sphere is either a shrinking spherical cap or a totally geodesic sphere. Then we show that under certain pointwise curvature pinching condition the ancient solution in a hyperbolic space is a family of shrinking spheres. We also obtain a rigidity result for ancient solutions in a nonnegatively curved space form under an asymptotic integral curvature…
In this paper we generalize the neck-stability theorem of Kleiner-Lott to a special class of four-dimensional nonnegatively curved Type I $κ$-solutions, namely, those whose asymptotic shrinkers are the standard cylinder $\mathbb{S}^3\times\mathbb{R}$. We use this stability result to prove a rigidity theorem: if a four-dimensional Type I $κ$-solution with nonnegative curvature operator has the standard cylinder $\mathbb{S}^3\times\mathbb{R}$ as its asymptotic shrinker, then it is exactly the cylinder with its standard shrinking metric.
New Volume Comparison Results and Volume Growth Rigidity of Gradient Ricci Almost Solitons
本文针对以ρ(x)为Bakry-Émery Ricci曲率下界的完备流形建立新体积比较定理，将Cao与Zhou的结果推广到收缩梯度Ricci几乎孤子，得到梯度Ricci几乎孤子的体积刚性及极大体积增长刚性结果。
Liouville Rigidity and Universal Spacelikeness Estimates for a Lorentzian Prescribed Mean Curvature Equation
该研究针对洛伦兹型给定平均曲率方程的整体严格类空解，在不同维数和参数范围证明了刘维尔定理，建立了全局高度与洛伦兹因子界，还结合张量恒等式得到几何半空间刚性定理。
A note on Rigidity of Shrinking Gradient Ricci Solitons with Constant Scalar Curvature
本文研究了具有常标量曲率的收缩梯度Ricci溶剂的刚性性质，证明了其在特定条件下可视为有限商空间，扩展了Cheng-Zhou的结果到高维情况。
In their proof of the positive energy theorem, Schoen and Yau showed that every asymptotically flat spacelike hypersurface M of a Lorentzian manifold which is flat along M can be isometrically imbedded with its given second fundamental form into Minkowski spacetime as the graph of a function from R^n to R; in particular, M is diffeomorphic to R^n. In this short note, we give an alternative proof of this fact. The argument generalises to the asymptotically hyperbolic case, works in every dimension n, and does not need a spin structure.
The Rigidity Theorems for Self-Shrinkers in the Mean Curvature Flow
证明欧氏空间中自收缩超曲面的一个挤压定理，通过加权庞加莱不等式及相关条件得出\(S\equiv1\)等结论，推广并改进前人成果。
Let $F_n :(Σ, h_n) \to \mathbb C^2$ be a sequence of conformally immersed Lagrangian self-shrinkers with a uniform area upper bound to the mean curvature flow, and suppose that the sequence of metrics $\{h_n\}$ converges smoothly to a Riemannian metric $h$. We show that a subsequence of $\{F_n\}$ converges smoothly to a branched conformally immersed Lagrangian self-shrinker $F_\infty : (Σ, h)\to \mathbb C^2$. When the area bound is less than $16π$, the limit $F_\infty$ is an embedded torus. When the genus of $Σ$ is one, we can drop the assumption on convergence $h_n\to h$. When the genus of $Σ…
We study deformations of shrinking Ricci solitons on a compact manifold M, generalising the classical theory of deformations of Einstein metrics. Using appropriate notions of twisted slices S_f inside the space of all Riemannian metrics on M, we define the infinitesimal solitonic deformations and the local solitonic pre-moduli spaces. We prove the existence of a finite dimensional submanifold of S_f x C^infty(M), which contains the pre-moduli space of solitons around a fixed shrinking Ricci soliton as an analytic subset. We define solitonic rigidity and give criteria which imply it.
A Bernstein Theorem for the Self-Shrinking $J$-Equation and Some Generalizations
本文证明 $\mathbb{C}^n$ 上自收缩 $J$-方程的每个整体光滑多重次调和解必为二次多项式，去除了复 Hessian 的渐近下界假设，并推广到一大类完全非线性椭圆算子。
Self-shrinkers are hypersurfaces that shrink homothetically under mean curvature flow; these solitons model the singularities of the flow. It it presently known that an entire self-shrinking graph must be a hyperplane. In this paper we show that the hyperplane is rigid in an even stronger sense, namely: For $2 \leq n \leq 6$, any smooth, complete self-shrinker $Σ^n\subset\mathbf{R}^{n+1}$ that is graphical inside a large, but compact, set must be a hyperplane. In fact, this rigidity holds within a larger class of almost stable self-shrinkers. A key component of this paper is the procurement of…
On the Geometry of Complete Spacelike LW-Submanifolds in Locally Symmetric Semi-Riemannian Spaces
研究完全类空LW子流形在局部对称半黎曼空间中的几何特性，通过刚性结果和分析技术推导出关于曲率的尖锐不等式，并统一了已知的分类定理。
Rigidity for the logarithmic Sobolev inequality on complete metric measure spaces
本文研究了Bakry-Émery Ricci曲率有下界的完备度量测度空间上对数Sobolev不等式的刚性问题，使用不同于Ohta和Takatsu的方法证明了等号成立时空间分裂出Gaussian收缩孤立子。
A time-flat condition on spacelike 2-surfaces in spacetime is considered here. This condition is analogous to constant torsion condition for curves in three dimensional space and has been studied in [2, 4, 5, 12, 13]. In particular, any 2-surface in a static slice of a static spacetime is time-flat. In this article, we address the question in the title and prove several local and global rigidity theorems for such surfaces in the Minkowski spacetime.
This paper studies rigidity for immersed self-shrinkers of the mean curvature flow of surfaces in the three-dimensional Euclidean space $\mathbb{R}^3.$ We prove that an immersed self-shrinker with finite $L$-index must be proper and of finite topology. As one of consequences, there is no stable two-dimensional self-shrinker in $\mathbb{R}^3$ without assuming properness. We conclude the paper by giving an affirmative answer to a question of Mantegazza.
Rigidity of Ricci-Pinched Complete Self-Shrinkers in Arbitrary Codimension
本文证明任意余维数中满足 Ricci 下界条件的完备自收缩子必为线性子空间或圆球，并给出余维数二时的尖锐刚性定理。
Relative Liouville Rigidity for Lagrangian Self-Similar Submanifolds with Legendrian Boundary
研究单位球中具有勒让德边界的紧致拉格朗日子流形，证明在精确性或接触角余弦符号假设下，自相似子流形必为赤道n-盘，并将二维极小刚性定理推广到高维和拉格朗日自相似情形。
Extremal and weakly trapped submanifolds in Generalized Robertson-Walker spacetimes
该研究针对广义Robertson-Walker时空中任意余维的类空子流形，借助最大值原理在抛物性假设下推导其刚性性质，进而实现对该时空下极值与弱俘获子流形的刻画。
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