Simon Raulot
We revisit and generalize a recent result of Cederbaum [C2, C3] concerning the rigidity of the Schwarzschild manifold for spin manifolds. This includes the classical black hole uniqueness theorems [BM, GIS, Hw] as well as the more recent uniqueness theorems for pho-ton spheres [C1, CG1, CG2].
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In this article, we prove a rigidity theorem for isometric embeddings into the Schwarzschild manifold, by using the variational formula of quasi-local mass.
The Witten spinorial argument has been adapted in several works over the years to prove positivity of mass in the asymptotically AdS and asymptotically hyperbolic settings in arbitrary dimensions. In this paper we prove a scalar curvature rigidity result and a positive mass theorem for asymptotically hyperbolic manifolds that do not require a spin assumption. The positive mass theorem is reduced to the rigidity case by a deformation construction near the conformal boundary. The proof of the rigidity result is based on a study of minimizers of the BPS brane action.
Two rigidity results for surfaces in Schwarzschild spacetimes
该研究在微分几何领域，证明了Schwarzschild时空标准零超曲面上满足两类平均曲率型方程的曲面的刚性结果，关联共形几何经典定理。
Biharmonic Conformal Surfaces and Dirac Factorization I: Exact Spinorial Encoding and Scalar--Chiral Rigidity
本文研究双调和共形曲面，通过构造Laplace型算子将Ou方程组等价为单个Dirac方程，分类标量-手性分解，证明其刚性由Dirac判别式控制，为更广构造提供基线。
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.
Motivated by the rigidity case in the localized Riemannian Penrose inequality, we show that suitable singular metrics attaining the optimal value in the Riemannian Penrose inequality is necessarily smooth in properly specified coordinates. If applied to hypersurfaces enclosing the horizon in a spatial Schwarzschild manifold, the result gives the rigidity of isometric hypersurfaces with the same mean curvature.
Outward minimizing $p$-capacity, horizons, and Schwarzschild rigidity
本文证明在非负数量曲率且渐近平坦的完备流形上，ADM 质量受向外极小化 $p$-容量下确界控制，等号刻画 Schwarzschild 外部并揭示视界，容量间隙可检测视界。
Static Manifolds with Boundary and Rigidity of Scalar Curvature and Mean Curvature
研究有边紧流形上标量曲率与边界平均曲率的局部覆盖性质，探讨非典型域的几何特性及其与刚性定理和Schwarzschild光球的关系。
A rigidity theorem for asymptotically flat static manifolds and its applications
本文证明 n<8 渐近平坦静态流形在 Minkowski 型不等式取等时的 Schwarzschild 刚性，并用于得到 Bartnik 数据延拓及低能等势光子面的唯一性。
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 define a notion of extrinsic black hole in pure Lovelock gravity of degree $k$ which captures the essential features of the so-called Lovelock-Schwarzschild solutions, viewed as rotationally invariant hypersurfaces with null $2k$-mean curvature in Euclidean space $\mathbb R^{n+1}$, $2\leq 2k\leq n-1$. We then combine a regularity argument with a rigidity result by Araújo-Leite to prove, under a natural ellipticity condition, a global uniqueness theorem for this class of black holes. As a consequence we obtain, in the context of the corresponding Penrose inequality for graphs established by …
An Einstein manifold is called scalar curvature rigid if there are no compactly supported volume-preserving deformation of the metric which increase the scalar curvature. We give various characterizations of scalar curvature rigidity for open Einstein manifolds as well as for closed Einstein manifolds. As an application, we construct mass-decreasing perturbations of the Riemannian Schwarzschild metric and the Taub-Bolt metric.
We study rigidity of minimal two-spheres $Σ$ that locally maximize the Hawking mass on a Riemannian three-manifold with a positive lower bound on its scalar curvature. After assuming strict stability of $Σ$, we prove that a neighborhood of it in $M$ is isometric to one of the deSitter-Schwarzschild metrics on $(- ε,ε)\times Σ$. We also show that if $Σ$ is a critical point for the Hawking mass on the deSitter-Schwarzschild manifold $\mathbb{R}\times\Sph^2$ and can be written as a graph over a slice $Σ_r=\{r\}\times\mathbb{S}^2$, then $Σ$ itself must be a slice, and moreover that slices are inde…
This paper proves a positive energy-momentum theorem for oriented Riemannian 3-manifolds that are asymptotic to a standard hyperbolic slice in anti de Sitter space-time. Analogously to the original Witten's proof in the asymptotically flat case, this result relies on spinorial methods. We also give a rigidity theorem: if the energy-momentum is degenerate (in a certain sense) then our 3-manifold can be isometrically embedded in anti de Sitter.
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.
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。
Modified Hawking mass and rigidity of three-manifolds with boundary
本文研究带负标量曲率与平均凸边界的三维流形刚性，通过自由边界严格稳定极小圆盘的面积估计，证明局部最大化修正 Hawking 质量的圆盘使流形局部等距于半 anti-de Sitter–Schwarzschild 流形。
Rigidity of positive mass theorem with fast metric decay
研究欧几里得空间上非负标量曲率且无穷远处快速衰减的度量，通过证明四维及以上维度中衰减率超史瓦西度量的此类度量必平坦，补充相关工作，建立正质量定理在快速度量衰减下的刚性猜想。
Rigidity and compactness with constant mean curvature in warped product manifolds
本文证明了扭曲积流形Brendle类中具有常分布平均曲率的可矫正边界的刚性，并以此刻画了平均曲率分布收敛到常数的边界极限，该新结果依赖于分布CMC-刚性的全部效力。
Motivated by the quasi-local mass problem in general relativity, we study the rigidity of isometric immersions with the same mean curvature into a warped product space. As a corollary of our main result, two star-shaped hypersurfaces in a spatial Schwarzschild or AdS-Schwarzschild manifold with nonzero mass differ only by a rotation if they are isometric and have the same mean curvature. We also give similar results if the mean curvature condition is replaced by an $σ_2$-curvature condition.
On the Poincaré-Einstein manifolds with cylindrical conformal infinity
本文证明具有柱形共形无穷远的非正曲率Poincaré-Einstein流形必为双曲，并在圆因子长度足够大或Weyl曲率能量足够小时给出唯一性，通过建立新的ε-正则性实现。
Remarks on mean curvature flow solitons in warped products
本文研究一般黎曼流形和扭曲积中平均曲率流孤立子的分裂性与刚性，涵盖稳定性、高斯映射像区域及整体图等几何条件，重点关注常曲率与施瓦西型空间。
Several rigidity results are proved for critical points of natural Riemannian functionals on the space of metrics on 3-manifolds. Two of these results are as follows. Let (N, g) be a complete Riemannian 3-manifold, satisfying one of the following variational conditions: (i) (N, g) has non-negative scalar curvature and is a critical point for the L^2 norm of the full curvature R among compact perturbations of (N, g). (ii) (N, g) has non-negative scalar curvature, a free isometric S^1 action, and is a critical point of the L^2 norm of R among compact volume non-increasing perturbations of (N, g)…
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