Volodymyr Kiosak, Vladimir S. Matveev
We prove that every complete Einstein (Riemannian or pseudo-Riemannian) metric $g$ is geodesically rigid: if any other complete metric $\bar g$ has the same (unparametrized) geodesics with $g$, then the Levi-Civita connections of $g$ and $\bar g$ coincide.
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A Finsler metric is geodesically reversible if geodesics remain geodesics after a change of orientation. Asymmetric norms on vector spaces and Funk metrics in the interior of convex bodies are examples of geodesically reversible metrics that are not necessarily sums of reversible metrics and closed 1-forms. However, there seem to be few such examples in closed manifolds. In this paper the theory of volumes and areas on Finsler spaces is applied to establish a number of rigidity theorems which partially explain this paucity of examples. These rigidity results settle some hitherto unsolved cases…
We construct a large class of new singularity-free static Lorentzian four-dimensional solutions of the vacuum Einstein equations with a negative cosmological constant. The new families of metrics contain space-times with, or without, black hole regions. Two uniqueness results are also established.
We show that if two 4-dimensional metrics of arbitrary signature on one manifold are geodesically equivalent (i.e., have the same geodesics considered as unparameterized curves) and are solutions of the Einstein field equation with the same stress-energy tensor, then they are affinely equivalent or flat. Under the additional assumption that the metrics are complete or the manifold is closed, the result survives in all dimensions >2.
In this article we study self-gravitating static solutions of the Einstein-ScalarField system in arbitrary dimensions. We discuss the existence and the non-existence of geodesically complete solutions depending on the form of the scalar field potential $V(ϕ)$, and provide full global geometric estimates when the solutions exist. Our main results are summarised as follows. For the Klein-Gordon field, namely when $V(ϕ)=m^{2}|ϕ|^{2}$, it is proved that geodesically complete solutions have Ricci-flat spatial metric, have constant lapse and are vacuum, (that is $ϕ$ is constant and equal to zero if …
An affine manifold is said to be geodesically complete if all affine geodesics extend for all time. It is said to be affine Killing complete if the integral curves for any affine Killing vector field extend for all time. We use the solution space of the quasi-Einstein equation to examine these concepts in the setting of homogeneous affine surfaces.
We show that the recent work of Lee [23] implies existence of a large class of new singularity-free strictly static Lorentzian vacuum solutions of the Einstein equations with a negative cosmological constant. This holds in all space-time dimensions greater than or equal to four, and leads both to strictly static solutions and to black hole solutions. The construction allows in principle for metrics (whether black hole or not) with Yang-Mills-dilaton fields interacting with gravity through a Kaluza-Klein coupling.
Sobolev-to-Lipschitz property of geodesically complete spaces with curvature bounded from above
证明每个曲率上有界且测地完备的长度空间具有指数无穷的Sobolev-to-Lipschitz性质，即每个W^{1,∞}空间中的Sobolev映射有Lipschitz代表，且Lipschitz常数等于映射的无穷能量。
It is proved that the only geodesically complete stationary vacuum solution of the Einstein equations is the empty Minkowski space, or a quotient of it by a discrete group of isometries, generalizing a classical result of Lichnerowicz. In addition, we obtain an apriori bound on the curvature of stationary vacuum solutions away from the boundary or horizon.
On the causality of gravitational pp-wave spacetimes: A null version of Anderson's rigidity theorem
本文证明平面波是唯一区分且强因果的引力 pp-波时空，并据此证明安德森刚性定理的空版本：满足完备性与强因果等条件的 4 维 Ricci 平坦 Brinkmann 时空必为平面波时空的商。
We connect two a priori unrelated topics, theory of geodesically equivalent metrics in differential geometry, and theory of compatible infinite dimensional Poisson brackets of hydrodynamic type in mathematical physics. Namely, we prove that a pair of geodesically equivalent metrics such that one is flat produces a pair of such brackets. We construct Casimirs for these brackets and the corresponding commuting flows. There are two ways to produce a large family of compatible Poisson structures from a pair of geodesically equivalent metrics one of which is flat. One of these families is $(n+1)(n+…
We prove that any 4-dimensional geodesically complete spacetime with a timelike Killing field satisfying the vacuum Einstein field equation $Ric(g_{M})=λg_{M}$ with nonnegative cosmological constant $λ\geq 0$ is flat. When dim $\geq 5$, if the spacetime is assumed to be static additionally, we prove that its universal cover splits isometrically as a product of a Ricci flat Riemannian manifold and a real line.
In this paper we prove that a conformally compact Einstein manifold with the round sphere as its conformal infinity has to be the hyperbolic space. We do not assume the manifolds to be spin, but our approach relies on the positive mass theorem for asymptotic flat manifolds. The proof is based on understanding of positive eigenfunctions and compactifications obtained by positive eigenfunctions.
We classify all self-dual Einstein four-manifolds invariant under a principal action of the three-dimensional Heisenberg group with non-degenerate orbits. The metrics are explicit and we find, in particular, that the Einstein constant can take any value. Then we study when the corresponding (Riemannian or neutral-signature) metrics are (geodesically) complete. Finally, we exhibit the solutions of non-zero Ricci-curvature as different branches of one-loop deformed universal hypermultiplets in Riemannian and neutral signature.
We study the solvability of the equation for the smooth function F, H=-k F g, on a geodesically complete pseudo-Riemannian manifold (M,g), H being the covariant Hessian of F. A similar equation was considered by Obata and Gallot in the Riemannian case for positive values of the constant k; the result was that the manifold must be the canonical sphere. In this generalized setting we obtain a range of possibilities, depending on the sign of k, the signature of the metric and the value of a certain first integral of the equation: the manifold is shown to be of constant sectional curvature or a wa…
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.
Let (M,g) be a compact oriented Einstein 4-manifold. Write R-plus for the part of the curvature operator of g which acts on self-dual 2-forms. We prove that if R-plus is negative definite then g is locally rigid: any other Einstein metric near to g is isometric to it. This is a chiral generalisation of Koiso's Theorem, which proves local rigidity of Einstein metrics with negative sectional curvatures. Our hypotheses are roughly one half of Koiso's. Our proof uses a new variational description of Einstein 4-manifolds, as critical points of the so-called poure connection action S. The key step i…
Einstein Structure of Four-Manifolds
本文通过标准四维球面形变证明Einstein度量偏离即破坏Einstein条件，并基于自对偶结构不可约分解分析四维流形Einstein结构，进一步支持其刚性。
Rigidity of quasi-Einstein metrics: The incompressible case
研究向量场无散度且非零的拟爱因斯坦度量，闭流形上的刚性，发现其总容许由该向量场生成的单参数等距群，并给出几何与拓扑限制。
We call a metric quasi-Einstein if the $m$-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, which contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We study properties of quasi-Einstein metrics and prove several rigidity results. We also give a splitting theorem for some Kähler quasi-Einstein metrics.
This paper extends our earlier results to higher dimensions using a different approach, based on the rigidity of complex structures on certain domains.
In this paper, we consider some rigidity results for the Einstein metrics as the critical points of some known quadratic curvature functionals on complete manifolds, characterized by some point-wise inequalities. Moreover, we also provide rigidity results by the integral inequalities involving the Weyl curvature, the trace-less Ricci curvature and the Sobolev constant, accordingly.
In this paper, we prove some rigidity results for the Einstein metrics as the critical points of a family of known quadratic curvature functionals on closed manifolds, characterized by some point-wise inequalities. Moreover, we also provide a few rigidity results that involve the Weyl curvature, the trace-less Ricci curvature and the Yamabe invariant, accordingly.
We study rigidity results for the Einstein metrics as the critical points of a family of known quadratic curvature functionals involving the scalar curvature, the Ricci curvature and the Riemannian curvature tensor, characterized by some pointwise inequalities involving the Weyl curvature and the traceless Ricci curvature. Moreover, we also provide a few rigidity results for locally conformally flat critical 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.
Using the new diffeomorphism invariants of Seiberg and Witten, a uniqueness theorem is proved for Einstein metrics on compact quotients of irreducible 4-dimensional symmetric spaces of non-compact type. The proof also yields a Riemannian version of the Miyaoka-Yau inequality.
Rigidity of weighted Einstein smooth metric measure spaces
本文研究了加权爱因斯坦光滑度量测度空间的几何结构，给出了完整的局部分类，并证明了完备情形下流形要么是空间形式的加权类比，要么属于特定的爱因斯坦扭曲积族。
Static vacuum near horizon geometries are solutions $(M,g,X)$ of a certain quasi-Einstein equation on a closed manifold $M$, where $g$ is a Riemannian metric and $X$ is a closed 1-form. It is known that when the cosmological constant vanishes, there is rigidity: $X$ vanishes and consequently $g$ is Ricci flat. We study this form of rigidity for all signs of the cosmological constant. It has been asserted that this rigidity also holds when the cosmological constant is negative, but we exhibit a counter-example. We show that for negative cosmological constant if $X$ does not vanish identically, …
We derive some elliptic differential inequalities from the Weitzenböck formulas for the traceless Ricci tensor of a Kähler manifold with constant scalar curvature and the Bochner tensor of a Kähler-Einstein manifold respectively. Using elliptic estimates and maximum principle, some $L^p$ and $L^\infty $ pinching results are established to characterize Kähler-Einstein manifolds among Kähler manifolds with constant scalar curvature, and others are given to characterize complex space forms among Kähler-Einstein manifolds. Finally, these pinching results may be combined to characterize complex spa…
Rigidity of conformal submersions and quasi-Einstein manifolds
本文研究仿射子浸入和准爱因斯坦流形的刚性概念，通过子浸入技术证明了闭合准爱因斯坦流形在λ>0时的刚性结果，探讨了曲率条件对子浸入和相关流形刚性的影响。
In this paper we study warped product Einstein metrics over spaces with constant scalar curvature. We call such a manifold rigid if the universal cover of the base is Einstein or is isometric to a product of Einstein manifolds. When the base is three dimensional and the dimension of the fiber is greater than one we show that the space is always rigid. We also exhibit examples of solvable four dimensional Lie groups that can be used as the base space of non-rigid warped product Einstein metrics showing that the result is not true in dimension greater than three. We also give some further natura…
We prove that every Einstein metric on the unit ball B^4 of C^2, asymptotic to the Bergman metric, is equal to it up to a diffeomorphism. We need a solution of Seiberg--Witten equations in this infinite volume setting. Therefore, and more generally, if M^4 is a manifold with a CR-boundary at infinity, an adapted spinc-structure which has a non zero Kronheimer--Mrowka invariant and an asymptotically complex hyperbolic Einstein metric, we produce a solution of Seiberg--Witten equations with an strong exponential decay property.
Rigidity of complete non-compact generalized m-quasi-Einstein manifolds
研究常数量曲率$R \le 0$、孤子函数$\lambda > 0$且$m > 1$的完备非紧梯度广义$m$-拟爱因斯坦流形，通过引入加权函数$v = e^{-f/m}\lambda$并证明其次调和性，得到五个刚性结果，每个都迫使流形为欧几里得空间。
Rigidity of complete Kähler--Einstein metrics under cscK perturbations
本文研究完备非紧Kähler-Einstein流形上cscK度量的刚性，证明Bergman度量在常标量曲率下为Kähler-Einstein，并结合Cheng猜想解决给出有界严格伪凸域的球形刻画。
We establish codimension 4 regularity of noncollapsed sequences of metrics with bounds on natural generalizations of the Ricci tensor. We obtain a priori L2 curvature estimates on such spaces, with diffeomorphism finiteness results and rigidity theorems as corollaries.
In [7] Klainerman introduced the hyperboloidal method to prove the global existence results for nonlinear Klein-Gordon equations by using commuting vector fields. In this paper, we extend the hyperboloidal method from Minkowski space to Lorentzian spacetimes. This approach is developed in [14] for proving, under the maximal foliation gauge, the global nonlinear stability of Minkowski space for Einstein equations with massive scalar fields, which states that, the sufficiently small data in a compact domain, surrounded by a Schwarzschild metric, leads to a unique, globally hyperbolic, smooth and…
Gradient Einstein-type warped products: rigidity, existence and nonexistence results via a nonlinear PDE
本文确立了构造梯度 Einstein 型扭曲度量的充要条件，导出了一般 Lichnerowicz 方程并证明了非线性椭圆方程正解的梯度估计，进而给出了此类度量的非存在性、刚性结果及显式构造实例。
We re-visit the eigenvalue estimate of the Dirac operator on spin manifolds with boundary in terms of the first eigenvalues of conformal Laplace operator as well as the conformal mean curvature operator. These problems were studied earlier by Hijazi-Montiel-Zhang and Raulot and we re-prove them under weaker assumption that a boundary chirality operator exists. Moreover, on these spin manifolds with boundary, we show that any $C^{3,α}$ conformal compactification of some Poincare-Einstein metric must be the standard hemisphere when the first nonzero eigenvalue of the Dirac operator achieves its …
We prove that the bi-invariant Einstein metric on $SU_{2n+1}$ is isolated in the moduli space of Einstein metrics, even though it admits infinitesimal deformations. This gives a non-Kähler, non-product example of this phenomenon adding to the famous example of $\mathbb{CP}^{2n}\times\mathbb{CP}^{1}$ found by Koiso. We apply our methods to derive similar solitonic rigidity results for the Kähler--Einstein metrics on `odd' Grassmannians. We also make explicit a connection between non-integrable deformations and the dynamical instability of metrics under Ricci flow.
A rigidity theorem for Einstein $4$-manifolds with sectional curvature of a fixed sign, and its consequences
研究具有半定截面曲率的4维爱因斯坦度量，通过刻画满足特定逐点不等式的此类度量，得出一系列推论，包括在非正曲率闭爱因斯坦及凯勒 - 爱因斯坦4维流形研究中的应用，还得到了凯勒 - 爱因斯坦情形下的尖锐不等式。
The orthogonal decomposition of the Webster curvature provides us a way to characterize some canonical metrics on a pseudo-Hermitian manifold. We derive some subelliptic differential inequalities from the Weitzenböck formulas for the traceless pseudo-Hermitian Ricci tensor and the Chern-Moser tensor of Sasakian manifolds with constant pseudo-Hermitian scalar curvature and Sasakian pseudo-Einstein manifolds respectively. By means of either subelliptic estimates or maximum principle, some rigidity theorems are established to characterize Sasakian pseudo-Einstein manifolds among Sasakian manifold…
On stability and scalar curvature rigidity of quaternion-Kähler manifolds
本文证明负标量曲率的四元数-凯勒流形作为爱因斯坦流形稳定且标量曲率刚性，并推广到特殊和乐情形，同时构造正标量曲率的半稳定但非刚性的反例。
We investigate rigidity and stability properties of critical points of quadratic curvature functionals on the space of Riemannian metrics. We show it is possible to "gauge" the Euler-Lagrange equations, in a self-adjoint fashion, to become elliptic. Fredholm theory may then be used to describe local properties of the moduli space of critical metrics. We show a number of compact examples are infinitesimally rigid, and consequently, are isolated critical points in the space of unit-volume Riemannian metrics. We then give examples of critical metrics which are strict local minimizers (up to diffe…
Let $(X^{n},g_+) $ $(n\geq 3)$ be a Poincaré-Einstein manifold which is $C^{3,α}$ conformally compact with conformal infinity $(\partial X, [\hat{g}])$. On the conformal compactification $(\overline{X}, \bar g=ρ^2g_+)$ via some boundary defining function $ρ$, there are two types of Yamabe constants: $Y(\overline{X},\partial X,[\bar g])$ and $Q(\overline{X},\partial X,[\bar g])$. (See definitions (\ref{def.type1}) and (\ref{def.type2})). In \cite{GH}, Gursky and Han gave an inequality between $Y(\overline{X},\partial X,[\bar g])$ and $Y(\partial X,[\hat{g}])$. In this paper, we first show that …
The spectral rigidity of Ricci soliton and Einstein-type manifolds
本文针对谱几何经典问题，在梯度收缩Ricci孤立子与上同调爱因斯坦流形两类条件下，证明除零散未知情形外，单个p对应的p-拉普拉斯特征值可确定截面曲率常值性，且可将等谱条件放宽为近等谱版本。
In this paper we prove rigidity results on critical metrics for quadratic curvature functionals, involving the Ricci and the scalar curvature, on the space of Riemannian metrics with unit volume. It is well-known that Einstein metrics are always critical points. The purpose of this article is to show that, under some curvature conditions, a partial converse is true. In particular, for a class of quadratic curvature functionals, we prove that every critical metric with non-negative sectional curvature must be Einstein.
Conformal Kähler rigidity of Einstein four-manifolds
研究紧致连通定向爱因斯坦四维流形，通过结合勒布伦共形归一化及相关方程恒等式，证明在特定特征值条件下度量为共形凯勒且有正数量曲率，还得到对完备里奇平坦四维流形等的扩展及挤压定理，K3 曲面等表明假设精确。
In this paper, we study compact generalized $τ$-quasi Ricci-harmonic metrics. In the first part, we explore conditions under which generalized $τ$-quasi Ricci-harmonic metrics are harmonic-Einstein and give some characterization results for it. In the second part, we obtain some rigidity results for compact $(τ, ρ)$-quasi Ricci-harmonic metrics which are special case of generalized $τ$-quasi Ricci-harmonic metrics. In the third part, we shall give two gap theorems for compact $τ$-quasi Ricci-harmonic metrics by showing some necessary and sufficient conditions for the metrics to be harmonic-Ein…
Bergman--Einstein Rigidity for Hartogs Domains over Bounded Homogeneous Domains
本文证明了在覆盖有界同构域的Hartogs域上Bergman度量的刚性定理，展示了Bergman度量为Kähler-Einstein度量、同构、与单位球体双全纯等条件的等价性，回答了Yau的问题。
Rigidity of conformal metrics with constant sixth-order $Q$-curvature
本文证明：在正数量曲率的闭连通爱因斯坦流形上，任何共形且具有常六阶Q-曲率的光滑度量必为爱因斯坦度量，利用Obata型恒等式得出刚性结论。
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.
The "new positive energy conjecture" Horowitz and Myers (1999) probes a possible nonsupersymmetric AdS/CFT correspondence. We consider a version formulated for complete, asymptotically Poincaré-Einstein Riemannian metrics $(M,g)$ with bounded scalar curvature $R\ge -n(n-1)$. This version then asserts that any such $(M,g)$ must have mass not less than the mass $m_0$ of a metric $g_0$ induced on a time-symmetric slice of a certain AdS soliton spacetime. The conjecture remains unproved, having so far resisted standard techniques. Little is known other than that the conjecture is true for metrics …
Given a Riemannian spin^c manifold whose boundary is endowed with a Riemannian flow, we show that any solution of the basic Dirac equation satisfies an integral inequality depending on geometric quantities, such as the mean curvature and the O'Neill tensor. We then characterize the equality case of the inequality when the ambient manifold is a domain of a Kähler-Einstein manifold or a Riemannian product of a Kähler-Einstein manifold with R (or with the circle S^1).
Generalized Einstein Laurent polynomials, Toric Kähler-Einstein Rigidity, and Finite Exponential Families
本文研究正有限指数和对数满足广义爱因斯坦方程的条件，分类了二元洛朗多项式，证明了环面Kähler-Einstein流形的刚性，并刻画了有限指数族，解决了相关猜想。
Curvature-dimension condition, rigidity theorems and entropy differential inequalities on Riemannian manifolds
本文通过信息论方法研究黎曼流形上的曲率-维度条件、刚性定理和熵微分不等式，提供了新的等价特征化和流形特征化。
We study geodesically complete and locally compact Hadamard spaces X whose Tits boundary is a connected irreducible spherical building. We show that X is symmetric iff complete geodesics in X do not branch and a Euclidean building otherwise. Furthermore, every boundary equivalence (cone topology homeomorphism preserving the Tits metric) between two such spaces is induced by a homothety. As an application, we can extend the Mostow and Prasad rigidity theorems to compact singular (orbi)spaces of nonpositive curvature which are homotopy equivalent to a quotient of a symmetric space or Euclidean b…
Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow
本文研究了 Ricci 流下准 Einstein 度量的演化，通过曲率恒等式证明了某些闭合准 Einstein 流形的刚性。
In this paper, we investigate the geometry of Einstein-type equation on a Riemannian manifold, unifying various particular geometric structures recently studied in the literature, such as critical point equation and vacuum static equation. We show various rigidity results of Einstein-type manifolds under assumptions of several curvature conditions.
On the rigidity of the complex Grassmannians
本文研究复Grassmann流形上对称度量的无穷小Einstein形变到二阶的可积性，通过证明Koiso障碍多项式的非零性对其进行代数表征，并证明了当n为奇数时该度量在模空间中是孤立的。
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 Einstein Manifolds under Two Eigenvalue Conditions on the Curvature Operator of the Second Kind
本文证明了在第二类曲率算子两个特征值下界条件下，闭爱因斯坦流形（维数n≥6）的刚性定理，得出流形为平坦或球空间形式。
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