Hassan Boualem, Marc Herzlich
Any Kaehler metric on the ball which is strongly asymptotic to complex hyperbolic space and whose scalar curvature is no less than the one of the complex hyperbolic space must be isometrically biholomorphic to it. This result has been known for some time in odd complex dimension and we provide here a proof in even dimension.
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In this paper we obtain rigidity results and obstructions on the topology at infinity of translating solitons of the mean curvature flow in the Euclidean space. Our approach relies on the theory of f-minimal hypersurfaces.
In this paper, we prove a rigidity theorem of asymptotically hyperbolic manifolds only under the assumptions on curvature. Its proof is based on analyzing asymptotic structures of such manifolds at infinity and a volume comparison theorem.
In this paper we prove that under a lower bound on the Ricci curvature and an asymptotic assumption on the scalar curvature, a complete conformally compact manifold $(M^{n+1},g)$, with a pole $p$ and with the conformal infinity in the conformal class of the round sphere, has to be the hyperbolic space.
A rigidity result for weakly asymptotically hyperbolic manifolds with lower bounds on Ricci curvature is proved without assuming that the manifolds are spin. The argument makes use of a quasi-local mass characterization of Euclidean balls from \cite{Miao} \cite{S_T} and eigenfunction compactification ideas from \cite{Qing}.
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}.
Rigidity of Poincaré-Einstein manifolds with flat Euclidean conformal infinity
本文证明了共形无穷远为平坦欧氏空间的庞加莱-爱因斯坦流形的刚性定理，通过分析曲率张量在适配边界定义函数水平集上的传播完成证明，并给出非紧共形无穷远例子及与欧氏体积增长里奇平坦流形的类比。
Rigidity for inscribed radius estimate of asymptotically hyperbolic Einstein manifold
本文将紧流形内切半径上界及刚性结果推广到渐近双曲爱因斯坦流形，通过证明相对体积上界并结合已有工作得到刚性。
This paper generalizes a rigidity result of complex hyperbolic spaces by M. Herzlich. We prove that an almost Hermitian spin manifold $(M,g)$ of real dimension $4n+2$ which is strongly asymptotic to $\hyp{\C}^{2n+1}$ and satisfies a certain scalar curvature bound must be isometric to the complex hyperbolic space. The fact that we do not assume $g$ to be Kähler reflects in the inequality for the scalar curvature.
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.
In this paper, we give an optimal inequality relating the relative Yamabe invariant of a certain compactification of a conformally compact Poin-car{é}-Einstein manifold with the Yamabe invariant of its boundary at infinity. As an application, we obtain an elementary proof of the rigidity of the hyper-bolic space as the only conformally compact Poincar{é}-Einstein manifold with the round sphere as its conformal infinity.
Ricci flow and the scalar curvature rigidity of Einstein manifolds
本研究综述了爱因斯坦流形的线性稳定性、动力学稳定性与标量曲率刚性关联的最新成果，覆盖多类爱因斯坦流形并阐释其与正质量定理的关系。
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。
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。
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.
We use the notion of intrinsic flat distance to address the almost rigidity of the positive mass theorem for asymptotically hyperbolic manifolds. In particular, we prove that a sequence of spherically symmetric asymptotically hyperbolic manifolds satisfying the conditions of the positive mass theorem converges to hyperbolic space in the intrinsic flat sense, if the limit of the mass along the sequence is zero.
Rigidity results for Riemann solitons
该文研究黎曼孤立子的刚性问题，给出平凡孤立子的条件，证明非紧梯度黎曼孤立子的无穷远性态可使其成为爱因斯坦流形，并得到完备收缩/稳定梯度黎曼孤立子的标量曲率估计。
In this paper, we study some intrinsic characterization of conformally compact manifolds. We show that, if a complete Riemannian manifold admits an essential set and its curvature tends to -1 at infinity in certain rate, then it is conformally compactifiable and the compactified metrics can enjoy some regularity at infinity. As consequences we prove some rigidity theorems for complete manifolds whose curvature tends to the hyperbolic one in a rate greater than 2.
In the spirit of Otal and Croke, we prove that a negatively-curved asymptotically hyperbolic surface is boundary distance rigid, where the distance between two points on the boundary at infinity is defined by a renormalized quantity.
We establish the positive energy theorem and a Penrose-type inequality for 3-dimensional asymptotically hyperboloidal initial data sets with toroidal infinity, weakly trapped boundary, and satisfying the dominant energy condition. In the umbilic case, a rigidity statement is proven showing that the total energy vanishes precisely when the initial data manifold is isometric to a portion of the canonical slice of the associated Kottler spacetime. Furthermore, we provide a new proof of the recent rigidity theorems of Eichmair-Galloway-Mendes [10] in dimension 3, with weakened hypotheses in certai…
This paper develops a Carleman type estimate for immersed surface in Euclidean space at infinity. With this estimate, we obtain an unique continuation property for harmonic functions on immersed surfaces vanishing at infinity, which leads to rigidity results in geometry.
We consider the boundary rigidity problem for asymptotically hyperbolic manifolds. We show injectivity of the X-ray transform in several cases and consider the non-linear inverse problem which consists of recovering a metric from boundary measurements for the geodesic flow.
Spectral Positive Mass Theorem for Asymptotically Hyperbolic 3-manifolds with Toroidal Infinity
针对具有环面无穷远的渐近双曲3流形，定义了适应谱标量曲率的质量不变量，并在谱标量曲率下界条件下证明其正性，同时得到刚性定理和带宽估计。
Rigidity results for asymptotically locally hyperbolic manifolds with lower bounds on scalar curvature are proved using spinor methods related to the Witten proof of the positive mass theorem. The argument is based on a study of the Dirac operator defined with respect to the Killing connection. The existence of asymptotic Killing spinors is related to the spin structure on the end. The expression for the mass is calculated and proven to vanish for conformally compact Einstein manifolds with conformal boundary a spherical space form, giving rigidity. In the 4-dimensional case, the signature of …
Let (M, g) be an (n + 1)-dimensional asymptotically locally hyperbolic (ALH) manifold with a conformal compactification whose conformal infinity is ($\partial$M, [$γ$]). We will first observe that Ch(M, g) $\le$ n, where Ch(M, g) is the Cheeger constant of M. We then prove that, if the Ricci curvature of M is bounded from below by --n and its scalar curvature approaches --n(n+1) fast enough at infinity, then Ch(M, g) = n if and only Y($\partial$M, [$γ$]) $\ge$ 0, where Y($\partial$M, [$γ$]) denotes the Yamabe invariant of the conformal infinity. This gives an answer to a question raised by J. …
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.
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.
We investigate asymptotically flat manifolds with cone structure at infinity. We show that any such manifold M has a finite number of ends. For simply connected ends we classify all possible cones at infinity, except for the 4-dimensional case where it remains open if one of the theoretically possible cones can actually arise. This result yields in particular a complete classification of asymptotically flat manifolds with nonnegative curvature: The universal covering of an asymptotically flat manifold with nonnegative sectional curvature is isometric to a product of Euclidean space and an asym…
Curvature at the infinity of asymptotically flat Einstein manifold
研究渐近平坦爱因斯坦流形无穷远处的曲率，通过构造特定坐标，在曲率\(L^{m/2}\)范数有限时，可在其无穷远处定义外尔张量，并为任意维的ALE里奇平坦流形/轨形定义重整化体积。
Boundary Rigidity in a fixed conformal class for Asymptotically Hyperbolic Manifolds
研究渐近双曲流形中边界距离决定度量的问题，证明若两个共形渐近双曲度量（简单或负曲率）的标记重正化边界距离在共形无穷的某些代表下一致，则两度量相等。
We consider complete Riemannian manifolds which satisfy a weighted Poincarè inequality and have the Ricci curvature bounded below in terms of the weight function. When the weight function has a non-zero limit at infinity, the structure of this class of manifolds at infinity are studied and certain splitting result is obtained. Our result can be viewed as an improvement of Li-Wang's result in \cite{LW3}.
Positive mass and rigidity for asymptotically flat tangent bundles
针对切丛上慢衰减渐近平坦度量，证明了其正质量定理与刚性定理，确定了广义渐近质量由底流形几何数据决定。
A Rigidity theorem on conformally compact Einstein manifolds in high dimensions
本文在高维情况下建立了 conformally compact Einstein 流形的刚性定理，扩展了四维结果。
The spacetime $AdS_2 \times S^2$ is well known to arise as the 'near horizon' geometry of the extremal Reissner-Nordstrom solution, and for that reason it has been studied in connection with the AdS/CFT correspondence. Here we consider asymptotically $AdS_2 \times S^2$ spacetimes that obey the null energy condition (or a certain averaged version thereof). Supporting a conjectural viewpoint of Juan Maldacena, we show that any such spacetime must have a special geometry similar in various respects to $AdS_2 \times S^2$, and under certain circumstances must be isometric to $AdS_2 \times S^2$.
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。
Rigidity of ideal symmetric sets
本文利用无穷远处对径集，在哈达玛空间中刻画了高秩模型几何（黎曼对称空间、欧几里得建筑和乘积空间）的刚性特征。
Let $Γ$ be a non-uniform lattice in $PU(p,1)$ without torsion and with $p\geq2 $. We introduce the notion of volume for a representation $ρ:Γ\rightarrow PU(m,1)$ where $m \geq p$. We use this notion to generalize the Mostow--Prasad rigidity theorem. More precisely, we show that given a sequence of representations $ρ_n:Γ\rightarrow PU(m,1)$ such that $\lim_{n \to \infty} \text{Vol}(ρ_n) =\text{Vol}(M)$, then there must exist a sequence of elements $g_n \in PU(m,1)$ such that the representations $g_n \circ ρ_n \circ g_n^{-1}$ converge to a reducible representation $ρ_\infty$ which preserves a to…
Rigidity of Asymptotically Hyperboloidal Initial Data Sets with Vanishing Mass
该论文证明质量为零的渐近双曲初始数据集必须等距嵌入闵可夫斯基空间，排除了辐射构型，证明基于旋量衰减估计，适用于所有维度。
We prove the rigidity of positive mass theorem for asymptotically hyperbolic manifolds. Namely, if the mass equality holds, then the manifold is isometric to hyperbolic space. The result was previously proven for spin manifolds or under special asymptotics.
Positive mass theorems for singular asymptotically hyperbolic manifolds
本文通过分析奇异空间上的 Yamabe 方程，建立了具有任意端点的奇异渐近双曲流形的正质量定理，并推导了在光滑情形下也新颖的刚性结果。
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。
We show that at the level of formal expansions, any compact Riemannian manifold is the sphere at infinity of an asymptotically conical gradient expanding Ricci soliton.
On Rigidity of ALE vector bundles
本文研究ALE向量丛的拓扑刚性，证明只有2-球面、射影平面和开可缩流形允许ALE切丛，并探讨其他拓扑与几何刚性。
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.
Microlocal analysis near null infinity in asymptotically flat spacetimes
本文提出了一种分析渐近平坦时空波方程在null无穷邻域正则性和衰减性质的新方法，并证明了相关正则性和衰减估计。
Generalizing the result of Li and Tam for the hyperbolic spaces, we prove an existence theorem on the Dirichlet problem for harmonic maps with $C^1$ boundary conditions at infinity between asymptotically hyperbolic manifolds.
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。
Rigidity Theorems for Asymptotically Euclidean $Q$-singular Spaces
本文证明了渐近欧几里得 $Q$-奇异空间中 $J$-平坦流形的刚性定理，并展示了 $J$-张量在控制无穷远处衰减率方面的几何作用。
Conformal boundary rigidity from null geodesic travel times
研究渐近反德西特时空中零测地线从共形无穷远出发并返回的聚焦性质，证明若所有从同一点出发的测地线在无穷远对极点重聚，则时空共形于反德西特空间，并给出在零能量条件、静态或全局平稳情形下的证明。
It is well known that the spacetime $\text{AdS}_2\times S^2$ arises as the `near horizon' geometry of the extremal Reisser-Nordstrom solution, and for that reason it has been studied in connection with the AdS/CFT correspondence. Motivated by a conjectural viewpoint of Juan Maldacena, the authors in [4] studied the rigidity of asymptotically $\text{AdS}_2\times S^2$ spacetimes satisfying the null energy condition. In this paper, we take an entirely different and more general approach to the asymptotics based on the notion of conformal infinity. This involves a natural modification of the usual…
We prove a perturbative result concerning the uniqueness of Kerr-Newman family of black holes: given an asymptotically flat space-time with bifurcate horizons, if it agrees with a non-extremal Kerr-Newman space-time asymptotically flat at infinity and it is sufficiently close to the Kerr-Newman family, then the space-time must be one of the Kerr-Newman solutions. The closeness to the Kerr-Newman family is measured by the smallness of a pair of Mars-Simon type tensors, which were introduced by Wong in \cite{Wong_09} to detect the Kerr-Newmann family.
In this note we prove a global rigidity result for asymptotically flat, scalar flat Euclidean hypersurfaces with a minimal horizon lying in a hyperplane, under a natural ellipticity condition. As a consequence we obtain, in the context of the Riemannian Penrose conjecture, a local rigidity result for the family of exterior Schwarzschild solutions (viewed as graphs in Euclidean space).
In this note, we establish the dihedral rigidity phenomenon for a collection of parabolic polyhedrons enclosed by horospheres in hyperbolic manifolds, extending Gromov's comparison theory to metrics with negative scalar curvature lower bounds. Our result is a localization of the positive mass theorem for asymptotically hyperbolic manifolds. We also motivate and formulate some open questions concerning related rigidity phenomenon and convergence of metrics with scalar curvature lower bounds.
In the Cauchy problem for asymptotically flat vacuum data the solution-jets along the cylinder at space-like infinity develop in general logarithmic singularities at the critical sets at which the cylinder touches future/past null infinity. The tendency of these singularities to spread along the null generators of null infinity obstructs the development of a smooth conformal structure at null infinity. For the solution-jets arising from time reflection symmetric data to extend smoothly to the critical sets it is necessary that the Cotton tensor of the initial three-metric h satisfies a certain…
For a closed, strictly convex projective manifold of dimension $n\geq 3$ that admits a hyperbolic structure, we show that the ratio of Hilbert volume to hyperbolic volume is bounded below by a constant that depends only on dimension. We also show that for such spaces, if topological entropy of the geodesic flow goes to zero, the volume must go to infinity. These results follow from adapting Besson--Courtois--Gallot's entropy rigidity result to Hilbert geometries.
In this paper, we introduce a particularly nice family of locally CAT(-1) spaces, which we call hyperbolic P-manifolds. For $X^3$ a simple, thick hyperbolic P-manifold of dimension 3, we show that certain subsets of the boundary at infinity of the universal cover of $X^3$ are characterized topologically. Straightforward consequences include a version of Mostow rigidity, as well as quasi-isometric rigidity for these spaces.
This paper provides a classification result for gravitational instantons with cubic volume growth and cyclic fundamental group at infinity. It proves that a complete hyperkähler manifold asymptotic to a circle fibration over the Euclidean three-space is either the standard $\rl^3 \times \sph^1$ or a multi-Taub-NUT manifold. In particular, the underlying complex manifold is either $\cx \times \cx/\ir$ or a minimal resolution of a cyclic Kleinian singularity.
In this paper we obtain the asymptotic behavior of solutions of the Klein-Gordon equation on Lorentzian manifolds $(X^\circ,g)$ which are de Sitter-like at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces. Under global assumptions on the (null)bicharacteristic flow, namely that the boundary of the compactification X is a union of two disjoint manifolds, Y+ and Y-, and each bicharacteristic converges to one of these two manifolds as the parameter along the bicharacteristic goes to plus infinity, and to the ot…
Mass and rigidity in almost Kähler geometry
本文研究了近凯勒几何中质量与刚性的性质，推导了渐近局部欧几里得流形的ADM质量公式，并证明了正质量定理和彭罗斯不等式，同时探讨了近凯勒-爱因斯坦流形的刚性现象。
The volume entropy of a compact metric measure space is known to be the exponential growth rate of the measure lifted to its universal cover at infinity. For a compact Riemannian $n$-manifold with a negative lower Ricci curvature bound and a upper diameter bound, it was known that it admits an almost maximal volume entropy if and only if it is diffeomorphic and Gromov-Hausdorff close to a hyperbolic space form. We prove the quantitative rigidity of almost maximal volume entropy for $\operatorname{RCD}$-spaces with a negative lower Ricci curvature bound and Riemannian manifolds with a negative …
We derive geometric formulas for the mass of asymptotically hyperbolic manifolds using coordinate horospheres. As an application, we obtain a new rigidity result of hyperbolic space: if a complete asymptotically hyperbolic manifold has scalar curvature lower bound -n(n-1) and is isometric to hyperbolic space outside a coordinate horosphere, then the manifold is isometric to hyperbolic space. In addition, we apply our formula to investigate regions near infinity that do not contribute to the mass quantity, which leads to improved rigidity results of hyperbolic space.
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