Yu Yan
We give some uniform estimates for constant mean curvature solutions of the conformal vacuum Einstein constraint equations on compact manifolds. Existence of those solutions was given in a paper by J. Isenberg.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this short note we discuss some recent results about two-positive Ricci curvature and their applications to positive Einstein curvature.
This note is devoted to study the implications of nonpositive isotropic curvature and negative Ricci curvature for Einstein $4-$Manifolds.
In this paper, we classify $n$-dimensional ($n\geq 5$) vacuum static spaces with harmonic curvature, thus extending the $4$-dimensional work by Kim-Shin \cite{KS}. As a consequence, we provide new counterexamples to the Fischer-Marsden conjecture on compact vacuum static spaces.
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 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.
We prove that simply connected Einstein four-manifolds of positive scalar curvature are conformally Kähler if and only if the determinant of the self-dual Weyl curvature is positive.
We provide a classification of Einstein submanifolds in space forms with flat normal bundle and parallel mean curvature. This extends a previous result due to Dajczer and Tojeiro for isometric immersions of Riemannian manifolds with constant sectional curvature.
Let (M,g) be an Einstein manifold of dimension n \geq 4 with nonnegative isotropic curvature. We show that (M,g) is locally symmetric.
In this paper we extend the local scalar curvature rigidity result in [6] to a small domain on general vacuum static spaces, which confirms the interesting dichotomy of local surjectivity and local rigidity about the scalar curvature in general in the light of the paper [10]. We obtain the local scalar curvature rigidity of bounded domains in hyperbolic spaces. We also obtain the global scalar curvature rigidity for conformal deformations of metrics in the domains, where the lapse functions are positive, on vacuum static spaces with positive scalar curvature, and show such domains are maximal,…
This paper contains a classification of all 3-dimensional manifolds with constant scalar curvature $S \not= 0$ that carry a non-trivial solution of the Einstein-Dirac equation.
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.
Toric Einstein 4-manifolds with nonnegative sectional curvature
本文证明了4-流形上具有非负截面曲率的$T^2$-不变爱因斯坦度量是局部对称的。
We derive a local curvature estimate for four-dimensional stationary solutions to the inheriting Einstein-Maxwell-Klein-Gordon equations. In particular, it implies that any such stationary geodesically complete solution with vanishing Poynting vector and proper coupling constants (like dark energy) is flat. We also generalize the result to higher dimensions.
We first show that the connected sum along submanifolds introduced by the second author for compact initial data sets of the vacuum Einstein system can be adapted to the asymptotically Euclidean and to the asymptotically hyperbolic context. Then, we prove that in any case, and generically, the gluing procedure can be localized, in order to obtain new solutions which coincide with the original ones outside of a neighborhood of the gluing locus.
Patterson discussed the curvature identities on Riemannian manifolds in [14], and a curvature identity for any 6-dimensional Riemannian manifold was independently derived from the Chern-Gauss-Bonnet Theorem [8]. In this paper, we provide the explicit formulae of Patterson's curvature identity that holds on 5-dimensional and 6-dimensional Einstein manifolds. We confirm that the curvature identities on the Einstein manifold from the previous work [8] are the same as the curvature identities deduced from Patterson's result. We also provide examples that support the theorems.
In this paper, we obtain classification of four-dimensional Einstein manifolds with positive Ricci curvature and pinched sectional curvature. In particular, the first result concerns with an upper bound of sectional curvature, improving a theorem of E. Costa. The second is a generalization of D. Yang's result assuming an upper bound on the difference between sectional curvatures.
We construct new homogeneous Einstein spaces with negative Ricci curvature in two ways: First, we give a method for classifying and constructing a class of rank one Einstein solvmanifolds whose derived algebras are two-step nilpotent. As an application, we describe an explicit continuous family of ten-dimensional Einstein manifolds with a two-dimensional parameter space, including a continuous subfamily of manifolds with negative sectional curvature. Secondly, we obtain new examples of non-symmetric Einstein solvmanifolds by modifying the algebraic structure of non-compact irreducible symmetri…
For an asymptotically Poincare-Einstein manifold with a lower Ricci curvature bound, we establish a sharp inequality relating the type II Yamabe invariant of the interior and the Yamabe invariant of its conformal infinity
Let (M,g) be a compact oriented Einstein 4-manifold. If M has positive intersection form and g has non-negative sectional curvature, we show that, up to rescaling and isometry, (M,g) is CP2, equipped with its standard Fubini-Study metric.
We prove that a simpy connected Hermitian Einstein 4-manifold with non-negative sectional curvature is isometric to complex projective space $\mathbb{C}\mathbb{P}^{2}$ with the Fubini-Study metric or isometric to the product $\mathbb{S}^{2}\times \mathbb{S}^{2}$ with the canonical metric.
Let C be a cone in the space of algebraic curvature tensors. Moreover, let (M,g) be a compact Einstein manifold with the property that the curvature tensor of (M,g) lies in the cone C at each point on M. We show that (M,g) has constant sectional curvature if the cone C satisfies certain structure conditions.
On quasi-Einstein manifolds with constant scalar curvature
本文分类了具有常数量曲率的$T$-平坦及三维$m$-拟爱因斯坦流形，并构造了新的非紧致例子。
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。
A Poincaré-Einstein metric $g$ is called non-degenerate if there are no non-zero infinitesimal Einstein deformations of $g$, in Bianchi gauge, that lie in $L^2$. We prove that a 4-dimensional Poincaré-Einstein metric is non-degenerate if it satisfies a certain chiral curvature inequality. Write $R_+$ for the part of the curvature operator of g which acts on self-dual 2-forms. We prove that if $R_+$ is negative definite then $g$ is non-degenerate. This is a chiral generalisation of a result due to Biquard and Lee, that a Poincaré-Einstein metric of negative sectional curvature is non-degenerate
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 study Riemannian functionals defined by $L^2$-norms of Ricci curvature, scalar curvature, Weyl curvature, and Riemannian curvature. We try to understand stability of their critical points that are products of Einstein metrics. In particular, we prove that the product of a spherical space form and a compact hyperbolic manifold is unstable for some quadratic functionals if the first eigenvalue of the Laplacian of the hyperbolic manifold is sufficiently small.
We establish variational formulas for Ricci upper and lower bounds, as well as a derivative formula for the Ricci curvature. As applications, constant curvature manifolds, Einstein manifolds and Ricci parallel manifolds are identified, respectively, with different integral-differential formulas and semigroup inequalities. Moreover, by using derivative and Hessian formulas for the heat semigroup $P_t$ developed from stochastic analysis, explicit Hessian estimates are derived on Einstein and Ricci parallel manifolds.
Inspired by the study of $V$-static manifold about classification, in this article, we apply the recent results obtained by Freitas and Gomes (Compact gradient Einstein-type manifolds with boundary, 2022) to prove the rigidity results for compact gradient Einstein-type manifolds with nonempty boundary and constant scalar curvature under some suitable pinching conditions. As a special case of gradient Einstein-type manifold, we also give a rigidity result of $(m,ρ)$-quasi-Einstein manifold with boundary.
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.
For Einstein four-manifolds with positive scalar curvature, we derive relations among various positivity conditions on the curvature tensor, some of which are of great importance in the study of the Ricci flow. These relations suggest possible new ideas to study the well-known rigidity conjecture for positively curved Einstein four-manifolds.
In this paper we will prove that the only compact 4-manifold M with an Einstein metric of positive sectional curvature which is also hermitian with respect to some complex structure on M, is the complex projective plane CP^2, with its Fubini-Study metric.
An $(α,β)$-metric is defined by a Riemannian metric $α$ and $1$-form $β$. In this paper, we study a known class of two-dimensional $(α,β)$-metrics of vanishing S-curvature. We determine the local structure of those metrics and show that those metrics are Einsteinian (equivalently, isotropic flag curvature) but generally are not Ricci-flat.
Einstein submanifolds with parallel mean curvature into $\mathbb{S}^n\times\mathbb{R}$
本文证明 $\mathbb{S}^n\times\mathbb{R}$ 中满足平坦法丛与平行平均曲率条件的 Einstein 子流形可表示为等距浸入的扭曲积。
Einstein manifolds and curvature operator of the second kind
本文利用 Bochner 技巧证明具有非负或一定阶数非负第二类曲率算子的紧致爱因斯坦流形必为常曲率空间，并结合已有结果给出统一判定条件。
Prescribed Ricci curvature near an Einstein manifold with boundary
该研究针对带边界的紧致爱因斯坦黎曼流形，证明了关联度量与其Ricci曲率、边界诱导共形类、边界平均曲率的映射在基准度量附近局部可逆，同时研究了反变Ricci算子及爱因斯坦算子等相关算子。
Vacuum static spaces with harmonic curvature
本文对具有调和曲率的真空静态空间进行了完整分类，证明其局部等距于四种类型之一，并由此得到完备情形的分类。
Building on previous results, we complete the classification of compact oriented Einstein 4-manifolds with det (W^+) > 0. There are, up to diffeomorphism, exactly 15 manifolds that carry such metrics, and, on each of these manifolds, such metrics sweep out exactly one connected component of the corresponding Einstein moduli space.
Bounds of Scalar curvature, S-curvature and distortion on $\infty$-Einstein Finsler manifolds
本文研究∞-Einstein Finsler流形的曲率与拓扑，通过对称条件构造特殊度量并建立S-曲率、畸变和标量曲率的界，得到拓扑有限性结论并部分回应Gromov猜想。
On Compact Quasi-Einstein Metrics of Constant Scalar Curvature
研究证明三维紧致常标量曲率准爱因斯坦度量为局部同构，并探讨其与Kähler-Einstein度量的关系
Pinched self-dual Weyl curvature on Einstein four-manifolds
研究Einstein四维流形中挤压自双曲Weyl曲率的性质，证明其在特定条件下具有反自双曲结构。
Let $(M^n,g)$, $n \ge 4$, be a compact simply-connected Riemannian manifold with nonnegative isotropic curvature. Given $0<l\le L$, we prove that there exists $\eps = \eps (l,L,n)$ satisfying the following: If the scalar curvature $s$ of $g$ satisfies $$ l \le s \le L $$ and the Einstein tensor satisfies $$ | Ric - \frac {s}{n}g | \le \eps$$ then $M$ is diffeomorphic to a symmetric space of compact type. This is a smooth analogue of the result of S. Brendle that a compact Einstein manifold with nonnegative isotropic curvature is isometric to a locally symmetric space.
Vacuum Static Spaces with Positive Isotropic Curvature
该研究刻画了具有正各向同性曲率的紧致真空静态空间，证明其在有限覆盖下只能是球面或圆周与低一维球面的乘积。
In this paper we first use the result in $[12]$ to remove the assumption of the $L^2$ boundedness of Weyl curvature in the gap theorem in $[9]$ and then obtain a gap theorem for a class of conformally compact Einstein manifolds with very large renormalized volume. We also uses the blow-up method to derive curvature estimates for conformally compact Einstein manifolds with large renormalized volume. The second part of this paper is on conformally compact Einstein manifolds with conformal infinities of large Yamabe constants. Based on the idea in $[15]$ we manage to give the complete proof of th…
Quasi-Einstein manifolds with Harmonic Weyl curvature
本文研究了具有调和韦伊尔曲率的准爱因斯坦流形，扩展了四维空间中的相关工作，并给出了新的非局部共形扁且非D-平坦的准爱因斯坦流形例子。
Given an Einstein structure with positive scalar curvature on a four-dimensional Riemannian manifolds, that is $Ric=λg$ for some positive constant $λ$. For convenience, the Ricci curvature is always normalized to $Ric=1$. A basic problem is to classify four-dimensional Einstein manifolds with positive or nonnegative curvature and $Ric=1$. In this paper, we firstly show that if the sectional curvature satisfies $K\le M_1= \frac{\sqrt{3}}2\approx 0.866025$, then the sectional curvature will be nonnegative. Next, we prove a family of rigidity theorems of Einstein four-manifolds with nonnegative s…
We introduce the notion of Kähler manifolds that are almost Einstein and we define a generalized mean curvature vector field along submanifolds in them. We prove that Lagrangian submanifolds remain Lagrangian, when deformed in direction of the generalized mean curvature vector field. For a Kähler manifold that is almost Einstein, and which in addition has a trivial canonical bundle, we show that the generalized mean curvature vector field of a Lagrangian submanifold is the dual vector field associated to the Lagrangian angle.
On a smooth metric measure spacetime $(M,g,e^{-f} dvol_g)$, we define a weighted Einstein tensor. It is given in terms of the Bakry-Émery Ricci tensor as a tensor which is symmetric, divergence-free, concomitant of the metric and the density function. We consider the associated vacuum weighted Einstein field equations and show that isotropic solutions have nilpotent Ricci operator. Moreover, the underlying manifold is a Brinkmann wave if it is $2$-step nilpotent and a Kundt spacetime if it is $3$-step nilpotent. More specific results are obtained in dimension $3$, where all isotropic solutions…
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…
Let $n \geq 3$ and $R_{abcd}$ be a $(4,0)$ sectionally positive curvature-type tensor (a tensor possessing all the local symmetries of the $(4,0)$ curvature tensor). Then there exists a metric tensor $g_{ab}$ such that $R_{abcd}\; g^{bd} = g_{ac} λ$ for some $λ$. Furthermore, $g_{ab}$ is unique up to a constant factor.
In this paper we show that a $ρ$-Einstein solitons conformal to a pseudo-Euclidean space, invariant under the action of the pseudo-orthogonal group with zero scalar curvature is stady and consequently flat. How application of the results obtained we present an explicit example for a the question proposed by Kazdan in [17].
Rigidity of positively curved vacuum Static spaces
本文研究维数≥4且具非负截面曲率的紧致真空静态空间，证明正截面曲率且Weyl曲率张量完全散度非负的这类空间等距于标准球面，还推广了临界点方程相关的刚性结果。
Let (M,g) be a 2-quasi-Einstein non-conformally flat semi-Riemannian manifold of dimension > 3. We prove that if its Riemann-Christoffel curvature tensor R is a linear combination of some Kulkarni-Nomizu tensors formed by the metric tensor g, the Ricci tensor S and its square S^2, then some pseudosymmetry type curvature conditions are satisfied. Certain non-conformally flat warped product manifolds with 2-dimensional base, and in particular some spacetimes, are such 2-quasi Einstein manifolds.
In this paper, we study the problem of conformally deforming a metric on a $3$-dimensional manifold $M^3$ such that its $k$-curvature equals to a prescribed function, where the $k$-curvature is defined by the $k$-th elementary symmetric function of the eigenvalues of the Einstein tensor, $1\le k\le 3$. We prove the solvability of the problem and the compactness of the solution sets on manifolds when $k=2$ and $3$, provided the conformal class admits a negative $k$-admissible metric with respect to the Einstein tensor.
Leon Green obtained remarkable rigidity results for manifolds of positive scalar curvature with large conjugate radius and/or injectivity radius. Using $C^{k,α}$ convergence techniques, we prove several differentiable stability and sphere theorem versions of these results and apply those also to the study of Einstein manifolds.
In this short note we are concerned with the Kahler-Einstein metrics near cone type log canonical singularities. By two different approaches, we construct a complete Kahler-Einstein metric with negative scalar curvature in a neighborhood of the cone over a Calabi-Yau manifold, which provides a local model for the future study of the global Kahler-Einstein metrics on singular varieties. In the first approach, we show that the singularity is uniformized by a complex ball and hence the induced metric from the Bergman metric of the ball is a desired one. In the second approach, we obtain a complet…
Weakly Einstein curvature tensors
本文对四维空间中的弱Einstein曲率张量进行了分类，揭示了其代数结构和几何应用。
Einstein manifolds of negative lower bounds on curvature operator of the second Kind
本文研究了具有负下界曲率运算符二类的爱因斯坦流形，证明其要么平坦要么为球面，并改进了经典结果。
Peng Wu recently announced a beautiful characterization of conformally Kaehler, Einstein metrics of positive scalar curvature on compact oriented 4-manifolds via the condition det (W^+) > 0. In this note, we buttress his claim by providing an entirely different proof of his result. We then present further consequences of our method, which builds on techniques previously developed in (LeBrun 2015).
Weighted Ricci curvatures with various weights of the S-curvature have played an important role in Finsler geometry, especially in the control of volumes of Finsler manifolds. In this paper, we study general weighted Ricci curvatures, and characterize Randers metrics of almost isotropic weighted Ricci curvatures.
Let Σbe a compact oriented surface immersed in a four dimensional Kähler-Einstein manifold M. We consider the evolution of Σin the direction of its mean curvature vector. It is proved that being symplectic is preserved along the flow and the flow does not develop type I singularity. When M has two parallel Kähler forms ω' and ω'' that determine different orientations andΣis symplectic with respect to both ω' and ω'', we prove the mean curvature flow of Σexists smoothly for all time. In the positive curvature case, the flow indeed converges at infinity.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。