The r-mean curvature and rigidity of compact hypersurfaces in the Euclidean space

空间形式中紧致双调和超曲面的刚性结果

Rigidity results for compact biconservative hypersurfaces in space forms

本文研究了空间形式中紧致双调和超曲面的刚性问题，提供了两个已知刚性结果的替代证明，并通过将非负截面曲率假设替换为形状算子平方范数的估计，证明了新的刚性结果。

常曲率空间中凸超曲面的局部刚性

Local rigidity of convex hypersurfaces in spaces of constant curvature

本文证明了常曲率空间（$n\ge4$）中凸等距超曲面在严格凸点附近局部全等，推广了Senkin在$C^1$光滑假设下的结果。

广义Robertson-Walker时空中随机完备极大超曲面的刚性结果

Rigidity results for stochastically complete maximal hypersurfaces in Generalized Robertson-Walker spacetimes

本文研究满足零能量条件的广义Robertson-Walker时空中随机完备极大超曲面，在适当几何假设下证明参数唯一性与不存在性结果，并建立Calabi-Bernstein型结论。

五维常曲率空间中极小旋转超曲面的一个刚性结果

A Rigidity Result for Minimal Rotation Hypersurfaces of 5D Spaces of Constant Curvature

本文证明五维常曲率空间中极小超曲面上的一个外在逐点不变量非正，且其恒为零当且仅当超曲面为旋转超曲面，并猜想该刚性结果可推广到其他维数。

Rigidity for nearly umbilical hypersurfaces in space forms

Perez proved some $L^2$ inequalities for closed convex hypersurfaces immersed in the Euclidean space $\mathbb{R}^{n+1}$, more generally, for closed hypersurfaces with non-negative Ricci curvature, immersed in an Einstein manifold. In this paper, we discuss the rigidity of these inequalities when the ambient manifold is $\mathbb{R}^{n+1}$, the hyperbolic space $\mathbb{H}^{n+1}$, or the closed hemisphere $\mathbb{S}_+^{n+1}$. We also obtain a generalization of the Perez's theorem to the hypersurfaces without the hypothesis of non-negative Ricci curvature.

Rigidity and bifurcation results for CMC hypersurfaces in warped product spaces

In this paper, we deduce some rigidity results in warped product spaces under normal variations of CMC hypersurfaces. In particular, we prove the existence of one-parameter families locally rigid on the spatial fiber of Anti-de Sitter Schwarzschild spacetime and one-parameter families with bifurcation points on the spatial fiber of de Sitter Schwarzschild spacetime.

关于在双曲空间中具有毛细边界紧致超曲面的一些刚性结果

Some rigidity results on compact hypersurfaces with capillary boundary in Hyperbolic space

本文证明了双曲空间中毛细超曲面的Heintze-Karcher不等式，仅在完全脐曲超曲面时成立，并应用该结果证明了嵌入毛细超曲面的Alexandrov型定理，还证明了在双曲空间中支撑于全测地平面的毛细超曲面的其他刚性结果。

高维黎曼流形中体积极小自由边界超曲面的刚性

Rigidity of volume-minimizing free boundary hypersurfaces in high dimensional Riemannian manifolds

本文针对高维黎曼流形中的体积极小自由边界超曲面，建立了基于数量曲率和边界第二基本形式的体积估计，并在等式情形下证明了流形的刚性。

On Rigidity of hypersurfaces with constant curvature functions in warped product manifolds

In this paper, we first investigate several rigidity problems for hypersurfaces in the warped product manifolds with constant linear combinations of higher order mean curvatures as well as "weighted'' mean curvatures, which extend the work \cite{Mon, Brendle,BE} considering constant mean curvature functions. Secondly, we obtain the rigidity results for hypersurfaces in the space forms with constant linear combinations of intrinsic Gauss-Bonnet curvatures $L_k$. To achieve this, we develop some new kind of Newton-Maclaurin type inequalities on $L_k$ which may have independent interest.

关于扭曲乘积流形中具有常移位曲率函数的超曲面的刚性

On rigidity of hypersurfaces with constant shifted curvature functions in warped product manifolds

本文推广了扭曲乘积流形中脐超曲面的刚性刻画，利用积分不等式与Minkowski型公式证明了常移位曲率函数超曲面在亚静态及一般扭曲乘积流形中的刚性定理。

Gap and rigidity theorems of $λ$-hypersurfaces

We study $λ$-hypersurfaces that are critical points of a Gaussian weighted area functional $\int_Σ e^{-\frac{|x|^2}{4}}dA$ for compact variations that preserve weighted volume. First, we prove various gap and rigidity theorems for complete $λ$-hypersurfaces in terms of the norm of the second fundamental form $|A|$. Second, we show that in one dimension, the only smooth complete and embedded $λ$-hypersurfaces in $\mathbb{R}^2$ with $λ\geq 0$ are lines and round circles. Moreover, we establish a Bernstein type theorem for $λ$-hypersurfaces which states that smooth $λ$-hypersurfaces that are enti…

A Rigidity Theorem for Hypersurfaces in Higher Dimensional Space Forms

The classical Cohn-Vossen theorem states that two isometric compact convex surfaces in $\mathbb{R}^{3}$ are congruent. In this short note, we generalize the classical Cohn-Vossen Theorem to higher dimensional surfaces in space form $N^{n+1}(K)$ for $n\ge 2$.

空间形式中常平均曲率超曲面的局部刚性（II）

Local rigidity of constant mean curvature hypersurfaces in space forms (II)

本文研究空间形式中常平均曲率超曲面的局部刚性，针对不同主曲率个数 $g\geq 4$ 的情形，在平均曲率恒定且存在足够多重主曲率时，证明其为等参超曲面。

欧几里得空间中严格凸超曲面与具有非零欧拉特征的非负曲率空间乘积的标量曲率刚性

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。

Rigidity of spacelike hypersurfaces in spatially weighted generalized Robertson-Walker spacetimes

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…

凸 NNSC 流形中完备自由边界极小超曲面的刚性

Rigidity of Complete Free Boundary Minimal Hypersurfaces in Convex NNSC Manifolds

本文证明了在 R^4 单位球中不存在完备的双侧稳定自由边界浸入，该结论源于利用翘曲 θ-气泡方法得出的在凸边界和非负中间 Ricci 曲率完备 4 维流形中极小超曲面的刚性定理。

空间形式中紧致双守恒超曲面：无标量曲率假设的刚性

Compact Biconservative Hypersurfaces in Space Forms: Rigidity Without Scalar Curvature Assumptions

本文研究空间形式中紧致双守恒超曲面的刚性，通过引入新的无散度张量，在不需要常标量曲率或截面曲率假设的条件下建立了刚性结果。

各向异性毛细超曲面的广义Minkowski公式与刚性结果

Generalized Minkowski formulas and rigidity results for anisotropic capillary hypersurfaces

该研究提出半空间中各向异性毛细超曲面的新Hsiung-Minkowski积分公式，证明相关Alexandrov型定理与刚性结果，得到各向异性Orlicz-Christoffel-Minkowski问题解的唯一性，为p≥1的L_p-Minkowski问题解的唯一性提供新证明。

Weighted Hsiung-Minkowski formulas and rigidity of umbilical hypersurfaces

We use the weighted Hsiung-Minkowski integral formulas and Brendle's inequality to show new rigidity results. First, we prove Alexandrov type results for closed embedded hypersurfaces with radially symmetric higher order mean curvature in a large class of Riemannian warped product manifolds, including the Schwarzschild and Reissner-Nordström spaces, where the Alexandrov reflection principle is not available. Second, we prove that, in Euclidean space, the only closed immersed self-expanding solitons to the weighted generalized inverse curvature flow of codimension one are round hyperspheres.

具有常曲率和交角条件的类空超曲面刚性

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。

欧氏球面中闭极小超曲面的强刚性

Strong Rigidity of Closed Minimal Hypersurfaces in Euclidean Spheres

本文证明了欧氏球面中闭嵌入极小超曲面的强刚性，并由此证实了 Choi-Schoen 猜想及 n<7 时的陈省身猜想。

关于双曲空间中具有常值位移曲率函数的超曲面的刚性

On rigidity of hypersurfaces with constant shifted curvature functions in hyperbolic space

本文研究双曲空间中具有常值位移曲率函数的超曲面刚性，通过常值加权位移平均曲率及其线性组合等条件给出测地球新刻画，并在更弱条件下推广了常值 Gauss-Bonnet 曲率 $L_k$ 的刚性结果。

Brown-York质量正性及其与Min-Oo猜想、Yau问题#100及超曲面刚性的紧致方法

A compact approach to the positivity of Brown-York's mass and its relation with the Min-Oo conjecture, Yau's Problem \#100 and rigidity of hypersurfaces

本文提出一种紧致方法探讨Brown-York质量的正性，并揭示其与Min-Oo猜想、Yau问题#100及超曲面刚性的内在联系。

Free boundary stable hypersurfaces in manifolds with density and rigidity results

Let $M$ be a weighted manifold with boundary $\partial M$, i.e., a Riemannian manifold where a density function is used to weight the Riemannian Hausdorff measures. In this paper we compute the first and the second variational formulas of the interior weighted area for deformations by hypersurfaces with boundary in $\partial M$. As a consequence, we obtain variational characterizations of critical points and second order minima of the weighted area with or without a volume constraint. Moreover, in the compact case, we obtain topological estimates and rigidity properties for free boundary stabl…

An intrinsic rigidity theorem for closed minimal hypersurfaces in 5-dimensional Euclidean sphere with constant nonnegative scalar curvature

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.

旋转支撑上极小毛细超曲面的Alexandrov型刚性

Alexandrov-Type Rigidity for Minimal Capillary Hypersurfaces on Rotational Supports

本文证明了旋转支撑上极小毛细超曲面的刚性，构造反例说明条件必要性，并给出钝角情形下的分类。

Rigidity of stable minimal hypersurfaces in asymptotically flat spaces

We prove that if an asymptotically Schwarzschildean 3-manifold (M,g) contains a properly embedded stable minimal surface, then it is isometric to the Euclidean space. This implies, for instance, that in presence of a positive ADM mass any sequence of solutions to the Plateau problem with diverging boundaries can never have uniform height bounds, even at a single point. An analogous result holds true up to ambient dimension seven provided polynomial volume growth on the hypersurface is assumed.

A Möbius scalar curvature rigidity on compact conformally flat hypersurfaces in $\mathbb{S}^{n+1}$

In this paper, we study conformally flat hypersurfaces of dimension $n(\geq 4)$ in $\mathbb{S}^{n+1}$ using the framework of Möbius geometry. First, we classify and explicitly express the conformally flat hypersurfaces of dimension $n(\geq 4)$ with constant Möbius scalar curvature under the Möbius transformation group of $\mathbb{S}^{n+1}$. Second, we prove that if the conformally flat hypersurface with constant Möbius scalar curvature $R$ is compact, then $$R=(n-1)(n-2)r^2, ~~0<r<1,$$ and the compact conformally flat hypersurface is Möbius equivalent to the torus $$\mathbb{ S}^1(\sqrt{1-r^2})…

5维和6维流形中常平均曲率超曲面的刚性

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。

Rigidity theorems for constant weighted mean curvature hypersurfaces

In this article, we study hypersurfaces $Σ\subset \mathbb{R}^{n+1}$ with constant weighted mean curvature. Recently, Wei-Peng proved a rigidity theorem for CWMC hypersurfaces that generalizes Le-Sesum classification theorem for self-shrinker. More specifically, they showed that a complete CWMC hypersurface with polynomial volume growth, bounded norm of the second fundamental form and that satisfies $|A|^2H(H-λ)\leq H^2/2$ must either be a hyperplane or a generalized cylinder. We generalize this result by removing the bound condition on the norm of the second fundamental form. Moreover, we prov…

A conformal differential invariant and the conformal rigidity of hypersurfaces

For a hypersurface V of a conformal space, we introduce a conformal differential invariant I = h^2/g, where g and h are the first and the second fundamental forms of V connected by the apolarity condition. This invariant is called the conformal quadratic element of V. The solution of the problem of conformal rigidity is presented in the framework of conformal differential geometry and connected with the conformal quadratic element of V. The main theorem states: Let n \geq 4 and V and V' be two nonisotropic hypersurfaces without umbilical points in a conformal space C^n or a pseudoconformal spa…

在R^{n+1}中具有常均曲率和常标量曲率的完备超曲面

Complete hypersurfaces in $R^{n+1}$ with constant mean and scalar curvature

本文研究了在R^{n+1}中具有常均曲率和常标量曲率的完备超曲面的刚性问题，提出了在不同维度下的刚性定理和特征化结果。

稳定极小超曲面的两个刚性结果

Two rigidity results for stable minimal hypersurfaces

本文证明了完备稳定极小超曲面的两个刚性结果：在$\mathbb{R}^4$中为超平面，且在$n\leq5$的正曲率闭Riemannian流形中不存在，均基于共形方法。

$\mathbb{S}^5$ 中闭极小超曲面的刚性

Rigidity of closed minimal hypersurface in $\mathbb{S}^5$

针对5维球面中第二基本形式平方长度恒定的闭浸入极小超曲面，本文在3-平均曲率和互异主曲率数均恒定的条件下证明其为等参超曲面，并限定 $S$ 只能取0、4或12，从而支持 Chern 猜想。

稳态空间中完备类空超曲面的刚性及不存在性

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 strictly convex domains in Euclidean spaces

In this paper, we prove a rigidity theorem for smooth strictly convex domains in Euclidean spaces.

Rigidity of volume-minimizing hypersurfaces in Riemannian 5-manifolds

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…

格林函数刚性与反演下超曲面的质量

Green function rigidity and the mass of hypersurfaces under inversion

本文在反演下证明了格林函数刚性猜想，针对 conformal Laplacian 和 Paneitz 运算符分别给出了不同维度下的结果，并推导了非整体图的 ADM 质量新公式。

四维超曲面在空间形式中的拓扑与刚性结果

Topological and rigidity results for four-dimensional hypersurfaces in space forms

本文研究四维超曲面在五维空间形式中的拓扑与刚性性质，通过Weyl张量的描述得到等参超曲面的新特征，推导闭合极小超曲面的Weyl函数拓扑界，并探讨第二基本形式的范数估计及刚性结果。

Explicit rigidity of almost-umbilical hypersurfaces

We give an explicit estimate of the distance of a closed, connected, oriented and immersed hypersurface of a space form to a geodesic sphere and show that the spherical closeness can be controlled by a power of an integral norm of the traceless second fundamental form, whenever the latter is sufficiently small. Furthermore we use the inverse mean curvature flow in the hyperbolic space to deduce the best possible order of decay in the class of $C^{\infty}$-bounded hypersurfaces of the Euclidean space.

Geometric Characterizations of Canal Hypersurfaces in Euclidean Spaces

In the present paper, firstly we obtain the general expression of canal hypersurfaces in Euclidean n-space and deal with canal hypersurfaces in Euclidean 4-space E4. We compute Gauss map, Gaussian curvature and mean curvature of canal hypersurfaces in E4 and obtain an important relation between the mean and Gaussian curvatures as 3Hrho = Krho^3-2. We prove that, the flat canal hypersurfaces in Euclidean 4-space are only circular hypercylinders or circular hypercones and minimal canal hypersurfaces are only generalized catenoids. Also, we state the expression of tubular hypersurfaces in Euclide…

德西特空间与闵可夫斯基空间中稳定类时毛细管超曲面的刚性

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。

共形结构与完备稳定极小超曲面的刚性

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。

H-hypersurfaces with at most 3 distinct principal curvatures in the Euclidean spaces

In this paper, we study hypersurfaces of Euclidean spaces with arbitrary dimension. First, we obtain some results on $\mbox{H}$-hypersurfaces. Then, we give the complete classification of $\mbox{H}$-hypersurfaces with 3 distinct curvatures. We also give explicit examples.

Warped product hypersurfaces in the pseudo-Euclidean space

We study hypersurfaces in the pseudo-Euclidean space $\mathbb{E}^{n+1}_s$, which write as a warped product of a $1$-dimensional base with an $(n-1)$-manifold of constant sectional curvature. We show that either they have constant sectional curvature or they are contained in a rotational hypersurface. Therefore, we first define rotational hypersurfaces in the pseudo-Euclidean space.

具有非负标量曲率的流形中极小超曲面的奇点去除刚性定理

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。

Anisotropic isoparametric hypersurfaces in Euclidean spaces

In this note, we give a classification of complete anisotropic isoparametric hypersurfaces, i.e., hypersurfaces with constant anisotropic principal curvatures, in Euclidean spaces, which is in analogue with the classical case for isoparametric hypersurfaces in Euclidean spaces. On the other hand, by an example of local anisotropic isoparametric surface constructed by B. Palmer, we find that anisotropic isoparametric hypersurfaces have both local and global aspects as in the theory of proper Dupin hypersurfaces.

Congruence of hypersurfaces of a pseudo-Euclidean space

Conditions, related to the so-called bending problem are considered for hypersurfaces of a pseudo-Euclidean space. Corresponding theorems are proved.

超曲面 Zoll 流形的拓扑与谱刚性

Topological and spectral rigidity of hypersurface Zoll manifolds

研究允许极小超曲面Zoll族的流形，证明其微分同胚于球面或实射影空间，并刻画Zoll族；进一步证明实射影三空间上的度量是曲面Zoll当且仅当射影宽度全相等，且闭曲面前四个宽度为2π时等距于常曲率一的实射影平面。

Generalized constant ratio hypersurfaces in Euclidean spaces

In this paper, we study generalized constant ratio (GCR) hypersurfaces in Euclidean spaces. We mainly focus on the hypersurfaces in $\mathbb E^4$. First, we deal with $δ(2)$-ideal GCR hypersurfaces. Then, we study on hypersurfaces with constant (first) mean curvature. Finally, we obtain the complete classification of GCR hypersurfaces with vanishing Gauss-Kronecker curvature. We also give some explicit examples. Keywords: Generalized constant ratio submanifolds, $δ(r)$-invariant hypersurfaces, constant mean curvature, Gauss-Kronecker curvature

A new gap for $CMC$ biharmonic hypersurfaces in Euclidean spheres

In this note we improve a gap result concerning the range of the mean curvature of complete $CMC$ proper-biharmonic hypersurfaces in unit Euclidean spheres.

Rigidity theorems of spacelike entire self-shrinking graphs in the pseudo-Euclidean space

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.

Translation hypersurfaces with constant Sr curvature in the Euclidean space

In this paper, we give a complete description of all translation hypersurfaces with constant r-curvature Sr, in the Euclidean space.

实空间形式中超曲面的极小性与非线性谱刚性

Minimality and Nonlinear Spectral Rigidity of Hypersurfaces in Real Space Forms

本文研究实空间形式中超曲面的非线性平均曲率谱方程，证明谱参数满足特定条件时的极小性及定量局部刚性，涵盖Chen猜想与双调和极小性情形。