Rigidity of Convex Surfaces in Homogeneous Spaces
Harold Rosenberg, Renato Tribuzy
本文证明了具有相同外曲率的齐次三维流形E（k; τ）（不同于空间型）中完备曲面的定向等距浸入的刚性。
We prove rigidity of oriented isometric immersions of complete surfaces in the homo- geneous 3-manifolds E(k; τ) (different from the space forms) having the same positive extrinsic curvature.
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A splitting of the local rigidity of Clifford-Klein forms of homogeneous spaces of completely solvable Lie groups
本文讨论了1-连通完全可解李群的齐性空间的Clifford-Klein型的局部刚性。实际上，我们引入了局部刚度的分解：竖向刚度和水平刚度。通过使用这种分裂，我们改进了一些现有的结果的局部刚性和介绍了一个新的途径Baklingo的猜想的局部刚性。
In this paper, we prove a rigidity theorem for smooth strictly convex domains in Euclidean spaces.
Rigidity of convex hypersurfaces, Rigidity of convex hypersurfaces in multidimensional spaces of constant curvature
本文移除了Senkin推广Pogorelov刚性定理时的超曲面光滑性假设，建立了高维欧氏空间及其他常曲率模型空间中任意紧凸超曲面的内蕴度量刚性结果。
Dihedral Rigidity for Convex Polytopes by Smooth Approximation
该研究遵循Brendle的光滑逼近方法，证明了凸多面体的Gromov二面刚性猜想
In this work we determine bifurcation instants for 1-parameter families of solutions to the Yamabe problem defined on maximal flag manifolds. We also study the local rigidity points, namely, a isolated solution of the Yamabe problem.
Scalar curvature rigidity of convex polytopes
本文证明了凸多面体的标量曲率刚性定理，方法上结合带边流形上Dirac算子的Fredholm理论与Fefferman-Phong型定理。
Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on spheres
利用变分方法研究了球面上常数量曲率度量共形和任意接近齐次度量的存在性和不存在性。这描述了所有的临界点的希尔伯特-爱因斯坦功能的共形类，近齐次度量。对于U（n+1），Sp（n+1）和Spin（9）-齐次度规的单参数族，得到了分支和局部刚性现象.
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。
We prove structure results for homogeneous spaces that support a non-constant solution to two general classes of equations involving the Hessian of a function and an invariant 2-tensor. We also consider trace-free versions of these systems. Our results generalize earlier rigidity results for gradient Ricci solitons and warped product Einstein metrics. In particular, our results apply to homogeneous gradient solitons of any invariant curvature flow and give a new structure result for homogeneous conformally Einstein metrics.
Any $6$-dimensional strict nearly Kähler manifold is Einstein with positive scalar curvature. We compute the coindex of the metric with respect to the Einstein-Hilbert functional on each of the compact homogeneous examples. Moreover, we show that the infinitesimal Einstein deformations on $F_{1,2}=\mathrm{SU}(3)/T^2$ are not integrable into a curve of Einstein metrics.
On the local rigidity of Einstein manifolds with convex boundary
设（M，g）是具有非空边界的紧Einstein流形.证明了当边界弱凸且基本群上的一个简单条件成立时，边界上的Killing场扩张到任意（M，g）的Killing场.这给出了欧几里得空间中凸曲面的经典无穷小刚性的一个新的证明，并将结果推广到任意维的Einstein度量。
Rigidity of Entire Convex Self-Shrinking Solutions to Hessian Quotient Flows
证明了Hessian商流在$\mathbb{R}^n$上的所有光滑严格凸自缩解都是二次的.推广了丁辛\cite{DX}关于伪欧几里得空间中拉格朗日中曲率流的整体自缩解的刚性定理。此外，我们表明，我们的论点适用于更大的一类方程。特别地，在一定条件下，我们得到了Kähler-里奇流在$\mathbb{C}^n$上的整体自缩解的刚性结果.
Bifurcation and Local Rigidity of Homogeneous Solutions to the Yamabe Problem on Aloff-Wallach Spaces
本文通过构造Aloff-Wallach空间上的Yamabe问题单参数解族并考察Morse指数变化确定分岔时刻，同时定义并区分了分岔点与局部刚性点。
On the infinitesimal rigidity of weakly convex polyhedra
这里的主要动机是一个问题：是否任何多面体，可以细分成凸片，而不增加一个顶点，并具有相同的顶点作为一个凸多面体，是无穷小刚性。我们证明，这确实是两类多面体的情况下：那些从一个凸多面体的“凹痕”在一个共同的顶点，并与一个自然的细分悬挂最多两个边缘。
Gromov asked if the bi-invariant metrics on a compact Lie group are extremal compared to any other metrics. In this note, we prove that the bi-invariant metrics on a compact connected semi-simple Lie group $G$ are extremal (in fact rigid) in the sense of Gromov when compared to the left-invariant metrics. In fact the same result holds for a compact connected homogeneous manifold $G/H$ with $G$ compact connect and semi-simple.
Homogeneous structures and rigidity of isoparametric submanifolds in Hilbert space
本文研究了可分希尔伯特空间中秩至少为2的等参子流形，Heintze和Liu的一个结果证明了这类子流形是齐次的，并将这类子流形M和M中的点x与一个标准齐次结构（切空间上的某个双线性映射）联系起来.我们证明了齐次结构与第二基本型一起编码了关于M的所有信息，并由此推出了M完全由第二基本型及其共变导数决定的刚性结果，从而使得这类子流形可分类.作为一个重要的步骤，我们表明，一个参数组的等距构造的Heintze和刘证明其同质性诱导光滑，因此无处不在定义的Killing场，这意味着连续性的均匀结构。这里一个重要的工具是引进仿射根系的等参子流形。
Rigidity of compact pseudo-Riemannian homogeneous spaces for solvable Lie groups
设$M$是紧致连通伪黎曼流形，其上可解的等距连通李群$G$作用传递.我们证明了$G$几乎自由地作用在$M$上，并且$M$上的度量是由$G$上的双不变伪黎曼度量引起的。进一步证明了$M$的等距群的单位连通区与$G$一致.
Fubini-Griffiths-Harris rigidity of homogeneous varieties
每个均匀嵌入的齐次品种的投影刚性的上界的确定和一个新的，不变的Fubini形式的特征。
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。
Infinitesimal rigidity of convex surfaces through the second derivative of the Hilbert-Einstein functional II: Smooth case
给出了处处具有正高斯曲率的光滑闭曲面的无穷小刚性的一个新证明。我们使用一个重新表述，取代变形的嵌入变形的度量内的身体有界的表面。通过研究带边界项的希尔伯特-爱因斯坦泛函的导数得到了证明。这种方法是在某种意义上的双重证明高斯无穷小刚性，即刚性相对于高斯曲率参数化的高斯映射，通过研究衍生物的体积有界的表面。我们还记得，Blaschke的经典证明的无穷小刚性也涉及到高斯无穷小刚性，但在不同的方式：而Blaschke使用高斯刚性相同的表面，我们使用高斯刚性的极对偶。在球面和双曲-德西特空间中，希尔伯特-爱因斯坦泛函和体积之间以及两种刚性之间存在完美对偶。我们还指出了未来的研究方向，包括凸核的无穷小刚性的双曲3-流形。
Rigidity of entire self-shrinking solutions to Kähler-Ricci flow with strictly real convex potential
证明了具有严格实凸位势的Kähler-里奇流在$\mathbb{C}^n$上的自缩整体解必是二次的.同样的论证也给出了由Q所得到的伪欧几里得空间中拉格朗日中曲率流的整体自缩解的刚性的逐点证明。Ding和Y.L.欣此外，我们表明，我们的论点适用于更大的一类方程。
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。
Let $M$ be a pseudo-Hermitian homogeneous space of finite volume. We show that $M$ is compact and the identity component $G$ of the group of holomorphic isometries of $M$ is compact. If $M$ is simply connected, then even the full group of holomorphic isometries is compact. These results stem from a careful analysis of the Tits fibration of $M$, which is shown to have a torus as its fiber. The proof builds on foundational results on the automorphisms groups of compact almost pseudo-Hermitian homogeneous spaces. It is known that a compact homogeneous pseudo-Kähler manifold splits as a product of…
Rigidity of free boundary minimal disks in mean convex three-manifolds
本文研究平均凸、标量曲率有正下界的黎曼三维流形中局部最大化修正 Hawking 质量的自由边界极小圆盘，并在严格稳定条件下证明其邻域具有半 de Sitter–Schwarzschild 刚性。
A Note on Scalar curvature comparison rigidity for compact domains
本文借助 Brendle 边界条件下的调和旋量，将 Gromov 凸多面体标量曲率刚性猜想推广至任意凸 Riemann 多面体型区域，并得到与 Shi-Tam 结果平行的曲率比较刚性定理。
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。
In this paper, we study Lagrangian submanifolds of the homogeneous nearly Kähler $6$-dimensional unit sphere $\mathbb{S}^6(1)$. As the main result, we derive a Simons' type integral inequality in terms of the second fundamental form for compact Lagrangian submanifolds of $\mathbb{S}^6(1)$. Moreover, we show that the equality sign occurs if and only if the Lagrangian submanifold is either the totally geodesic $\mathbb{S}^3(1)$ or the Dillen-Verstraelen-Vrancken's Berger sphere $S^3$ discribed in J Math Soc Japan, 42: 565-584, 1990.
On the rigidity of mean convex self-shrinkers
自收缩模型的奇异性的中曲率流，他们被定义为特殊的解决方案，合同下的流动相似。Colding-Ilmanen-Minicozzi证明了圆柱形的自收缩体在很强的意义上是刚性的--也就是说，任何自收缩体都是凸的，在一个大的但紧凑的集合上具有一致的有界曲率，它必须是一个圆柱体。利用这个结果，Colding和Minicozzi能够建立在圆柱奇点处爆破的唯一性，并提供了一般中曲率流的奇异集的详细描述。在本文中，我们表明，有界曲率假设是不必要的圆柱的刚度，如果n是最多6，或如果中曲率是有界以下由一个正常数。这些结果遵循曲率估计，我们证明严格平均凸自收缩。我们还得到了一个刚性定理在所有尺寸的图形自收缩，和曲率估计的翻译的中曲率流。
A higher rank rigidity theorem for convex real projective manifolds
本文研究凸实射影流形的刚性问题，将 Ballmann 与 Burns-Spatzier 的高阶刚性定理推广到该情形，证明了相应类似定理。
We classify smooth Schubert varieties S_0 in a rational homogeneous manifold S associated to a short root, and show that they are rigid in the sense that any subvariety of S having the same homology class as S_0 is induced by the action of Aut_0(S), unless S_0 is linear.
Compact approach to the positivity of Brown-York mass and rigidity of manifolds with mean-convex boundaries in flat and spherical contexts
本文利用旋量证明与 ADM 质量正定性，为 Brown-York 质量正定性提供了紧致方法，并得出 Dirac 算子特征值下界及平坦与球面空间中平均凸流形的刚性结果。
We prove entropy rigidity for finite volume strictly convex projective manifolds in dimensions $\geq 3$, generalizing the work of arXiv:1708.03983 to the finite volume setting. The rigidity theorem uses the techniques of Besson, Courtois, and Gallot's entropy rigidity theorem. It implies uniform lower bounds on the volume of any finite volume strictly convex projective manifold in dimensions $\geq 3$.
Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane
本文研究常曲率曲面上凸台球的Hopf型刚性。证明了在双曲平面或半球上唯一无共轭点的凸台球是圆台球。
Rigidity of invariant convex sets in symmetric spaces
主要结果是：不可约高阶对称空间的真凸子集不可能有Zebraki稠密稳定子。
On the rigidity of admissible pairs of rational homogeneous spaces of Picard number one which are of deletion type
本文应用Mok的一般性判据，给出删除型不可约紧致Hermitian对称空间的可容许对在有理饱和条件下具有刚性的具体例子。
Scalar curvature rigidity of parabolic convex polytopes in hyperbolic space
本文利用旋量技术与Brendle-Wang平滑构造，证明了双曲空间中抛物凸多面体的数量曲率刚性，并进一步得出有界光滑区域的Llarull型刚性与多面体初始数据集的二面角刚性。
Deformations of hyperbolic convex polyhedra and 3-cone-manifolds
斯托克问题，首次制定于1968年，在于理解在何种程度上一个凸多面体是由它的二面角。通过二重构造，这个问题与三维锥流形的刚性问题密切相关。在以前的论文中，证明了两个这样的刚性结果，这意味着无穷小版本的斯托克猜想在双曲和欧几里得情况下是正确的。在第二篇文章中，我们证明了局部刚性成立，并得到了具有给定组合类型的凸双曲多面体空间局部参数化于二面角的集合，以及双曲3-锥流形的类似陈述。
Rigidity of convex divisible domains in flag manifolds
与实射影空间中凸可分域的许多例子不同，我们证明了在射影同构下，当$p > 1$满足时，在$\mathbb{R}^{2p}$中的$p$ -平面的Grassmannian中只有一个凸可分域.此外，这个凸可分区域是与单李群SO $(p, p)$相关联的对称空间的模型。
Rigidity of area-minimizing free boundary surfaces in mean convex three-manifolds
本文证明了具有平均凸边界和下有界的数量曲率且含有局部面积极小化的自由边界曲面的三流形的局部分裂定理。我们的方法基于Micallef和Moraru的方法。我们利用这个局部结果建立了面积最小化自由边界圆盘的整体刚性定理。在负数量曲率的情况下，这个整体结果意味着一个刚性定理的长度最小化边界的高原问题的解决方案。
Scalar curvature rigidity of parabolically convex domains in hyperbolic spaces
针对双曲空间中抛物凸域，证明了满足特定条件（映射度数非零、边界映射不增距离和平均曲率等）的自旋流形$N$是双曲的且边界与$M$边界等距，部分推广了Lott关于标量曲率刚性的结果至负下界情形。
Infinitesimal rigidity of convex surfaces through the second derivative of the Hilbert-Einstein functional I: Polyhedral case
摘要给出了凸多面体无穷小刚性的一个新证明。证明是基于研究衍生物的离散希尔伯特-爱因斯坦功能的空间上的“翘曲多面体”与一个固定的度量的边界。这种方法在某种意义上是双重的使用衍生物的体积，以证明高斯无穷小刚性的凸多面体，它处理的变形，保持面法线和面面积。在球面和双曲-德西特空间中，希尔伯特-爱因斯坦泛函和体积之间以及两种刚性之间存在完美对偶。我们还讨论了未来的研究方向，包括双曲（锥）流形的无穷小刚性的初等证明和离散Bochner技术的发展。
Isospectral simply-connected homogeneous spaces and the spectral rigidity of group actions
我们推广的Sunada的方法，以产生新的例子封闭，局部非等距流形是isospectral。特别是，我们生产对等谱，单连通，局部非等距正常齐性空间。这些对也让我们看到，一般来说，具有离散光谱的群作用并不取决于它们的光谱的可测量的共轭性。特别是，我们表明这对晶格行动。
Rigidity of complex convex divisible sets
如果存在一个共紧作用的离散投影自同构群，则称实射影空间中的开凸集是可除的。有许多这样的例子集和一个定理的Benoist意味着，许多这些例子是严格凸的，有$C^1$边界，并有字双曲除群。本文研究了复射影空间中的一个凸性概念，证明了具有$C^1$边界的可分复凸集是射影球。
We survey the main extensions of the classical Hadamard, Liebmann and Cohn-Vossen rigidity theorems on convex surfaces of $3$-Euclidean space to the context of convex hypersurfaces of Riemannian manifolds. The results we present include the one by Professor Renato Tribuzy (in collaboration with H. Rosenberg) on rigidity of convex surfaces of homogeneous $3$-manifolds.
In this paper we show that many projective Anosov representations act convex cocompactly on some properly convex domain in real projective space. In particular, if a non-elementary word hyperbolic group is not commensurable to a non-trivial free product or the fundamental group of a closed hyperbolic surface, then any projective Anosov representation of that group acts convex cocompactly on some properly convex domain in real projective space. We also show that if a projective Anosov representation preserves a properly convex domain, then it acts convex cocompactly on some (possibly different)…
Convex real projective structures and Weil's local rigidity Theorem
对于$n$维实双曲流形$M$，我们计算了特征簇$χ(π_1(M),SL(n+1,\mathbb R)), n>2$在Fuchisan轨迹上的Zerkki切空间，证明了该切空间由三次型组成.此外，我们还利用实射影结构证明了均匀双曲格的Weil局部刚性定理。
Modified support measures and Alexandrov-Fenchel type inequalities for nonsmooth h-convex bodies in hyperbolic space
该研究针对双曲空间中的h-凸体，通过球面卷积实现其光滑逼近，推广了两类Alexandrov-Fenchel不等式，引入曲率流证明其等号刚性，还建立了相关刚性定理以刻画测地球。
Rigidity and weak notions of spectral scalar curvature
本文通过二面角刚性结果，为谱纯量曲率从下方有界提出弱表述，并证明满足特定条件的度量必为平坦，受Gromov弱概念启发。
Rigidity properties of holomorphic isometries into homogeneous Kähler manifolds
本文证明了齐性凯勒流形上全纯等距映射的两个刚性定理，揭示特殊旗流形、平坦空间与齐性有界域在凯勒乘积下的非相对性，并推广了多项已有结果。
We show that a compact Riemannian $3$-manifold $M$ with strictly convex simply connected boundary and sectional curvature $K\leq a\leq 0$ is isometric to a convex domain in a complete simply connected space of constant curvature $a$, provided that $K\equiv a$ on planes tangent to the boundary of $M$. This yields a characterization of strictly convex surfaces with minimal total curvature in Cartan-Hadamard $3$-manifolds, and extends some rigidity results of Greene-Wu, Gromov, and Schroeder-Strake. Our proof is based on a recent comparison formula for total curvature of Riemannian hypersurfaces,…
Rigidity of Free Boundary Surfaces in Compact 3-Manifolds with Strictly Convex Boundary
本文对具有非负Ricci曲率和严格凸边界$\partial M$的紧致流形$M^3$，在3维空间中得到了Toponogov定理的一个类似结果.这里我们得到了$M^3$中自由边界极小曲面$Σ^2$的边界$\partialΣ$的长度$L(\partialΣ)$的一个精确上界，它是根据$Σ$的亏格和$\partialΣ$的连通分支数给出的，假设$Σ$的指数为1.之后，在一个自然的假设下的几何$M$沿$\partial M$，我们证明了，如果$L(\partialΣ)$饱和各自的上限，那么$M^3$是等距的欧几里德3-球和$Σ^2$是等距的欧几里德磁盘。特别地，当$M^3$是$\mathbb R^3$中的严格凸体时，我们得到了$Σ$的面积的一个精确上界.我们也考虑了类似的结果为固定稳定的表面。
We consider isometric immersions of complete connected Riemannian manifolds into space forms of nonzero constant curvature. We prove that if such an immersion is compact and has semi-definite second fundamental form, then it is an embedding with codimension one, its image bounds a convex set, and it is rigid. This result generalizes previous ones by M. do Carmo and E. Lima, as well as by M. do Carmo and F. Warner. It also settles affirmatively a conjecture by do Carmo and Warner. We establish a similar result for complete isometric immersions satisfying a stronger condition on the second funda…
Rigidity of smooth Schubert varieties in Hermitian symmetric spaces
Schubert簇是齐次流形G/P的不可约子簇，它对于理解齐次流形G/P的几何和半单李群G的作用有重要意义。考虑G/P中的有效圈空间，其同调类等于W型Schubert簇X_w的同调类的整数倍.本文证明了对于Hermitian对称空间中的光滑Schubert簇X_w，如果有平凡的例外，则这个循环空间是单的：任何具有同调类[X]=r[X_w]的不可约子簇X仍然是w型Schubert簇。
We study the boundary and lens rigidity problems on domains without assuming the convexity of the boundary. We show that such rigidities hold when the domain is a simply connected compact Riemannian surface without conjugate points. For the more general class of non-trapping compact Riemannian surfaces with no conjugate points, we show lens rigidity. We also prove the injectivity of the X-ray transform on tensors in a variety of settings with non-convex boundary and, in some situations, allowing a non-empty trapped set.
A compact Fuchsian manifold with boundary is a hyperbolic 3-manifold homeomorphic to $S_g \times [0; 1]$ such that the boundary component $S_g \times \{ 0\}$ is geodesic. We prove that a compact Fuchsian manifold with convex boundary is uniquely determined by the induced path metric on $S_g \times \{1\}$. We do not put further restrictions on the boundary except convexity.
It was proved in \cite{NS1} that obstacles $K$ in $\R^d$ that are finite disjoint unions of strictly convex domains with $C^3$ boundaries are uniquely determined by the travelling times of billiard trajectories in their exteriors and also by their so called scattering length spectra. However the case $d = 2$ is not properly covered in \cite{NS1}. In the present paper we give a separate different proof of the same result in the case $d = 2$.
Rigidity and regularity for almost homogeneous spaces with Ricci curvature bounds
本文研究几乎齐次RCD$(K,N)$空间的Gromov-Hausdorff极限，证明其必为Ricci曲率下界为$K$的幂零Lie群，并建立了拓扑刚性定理及光滑情形下Riemannian orbifold的刚性与$ε$-正则性结果。
Rigidity of Immersed Gauss Curvature Shrinkers in Euclidean Space
该研究证明，对任意$m\geq2$，满足$K=\langle X,\nu\rangle$的连通闭定向光滑浸入必为单位球面上的嵌入，通过分析极浸入与面积测度完成刚性分类。
We prove splitting theorems for mean convex open subsets in RCD (Riemannian curvature-dimension) spaces that extend results by Kasue, Croke and Kleiner for Riemannian manifolds with boundary to a non-smooth setting. A corollary is for instance Frankel's theorem. Then, we prove that our notion of mean curvature bounded from below for the boundary of an open subset is stable w.r.t. to uniform convergence of the corresponding boundary distance function. We apply this to prove almost rigidity theorems for uniform domains whose boundary has a lower mean curvature bound.
Rigidity Theorems for the Weyl Problem of Convex Surfaces in Hyperbolic 3-Space
研究双曲三维空间中非紧凸集的刚性，证明若两个非紧闭凸子集的理想边界为圆型闭集且连通分支可数，则边界间的内蕴等距可延拓为整体等距，推广了Luo-Luo-Rao的刚性定理，并建立了Weyl问题的唯一性结果。
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