Giovanni Catino
In this short note we prove that any complete four dimensional anti-self-dual (or self-dual) quasi-Einstein manifolds is either Einstein or locally conformally flat. This generalizes a recent result of X. Chen and Y. Wang.
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In this short note we discuss some recent results about two-positive Ricci curvature and their applications to positive Einstein curvature.
The aim of this note is to give an explicit description of quasi-Einstein metrics on $\Bbb{H}^{n}\times \Bbb{R}.$ We shall construct two examples of quasi-Einstein metrics on this manifold and then we shall prove the uniqueness of these examples. Finally, we shall describe the closed relation between quasi-Einstein metrics and static metrics in the quoted space.
In this note, we show that a nontrivial, compact, degenerate or nondegenerate, gradient Einstein-type manifold of constant scalar curvature is isometric to the standard sphere with a well defined potential function. Moreover, under some geometric assumptions, the noncompact case is also treated. In this case, the main result is that a homogeneous, proper, noncompact, nondegenerate, gradient Einstein-type manifold is an Einstein manifold.
In this paper, we prove some compactness theorems of Myers, Ambrose, and Galloway for complete Riemannian manifold in the concept of $h$-almost Ricci tensors and generalized quasi-Einstein tensors. Also, we extend the previous theorems when $h$ has at most linear growth in the distance function.
We study the volume of compact Riemannian manifolds which are Einstein with respect to a metric connection with (parallel) skew-torsion. We provide a result for the sign of the first variation of the volume in terms of the corresponding scalar curvature. This generalizes a result of M. Ville, related with the first variation of the volume on a compact Einstein manifold.
The aim of this note is the study of Einstein condition for para-holomorphic Riemannian metrics in the para-complex geometry framework. Firstly, we make some general considerations about para-complex Riemannian manifolds (not necessarily para-holomorphic). Next, using an one-to-one correspondence between para-holomorphic Riemannian metrics and para-Kähler Norden metrics, we study the Einstein condition for a para-holomorphic Riemannian metric and the associated real para-Kähler Norden metric on a para-complex manifold. Finally, it is shown that every semi-simple para-complex Lie group inherits…
A note on Einstein metrics and Riemannian twistor spaces
本文研究正标量曲率爱因斯坦四维流形的扭量刻画，证明扭量空间在纤维上标量曲率为常数的爱因斯坦四维流形必为半共形平坦，并分类了紧致正标量曲率情形及扭量空间 Ricci 平行的完备四维流形。
In this note, we construct new examples of Lorentzian Sasaki-Einstein (LSE) metrics on Smale manifolds $M.$ It has already been established in \cite{Gmz2} that such metrics exist on the so-called torsion free Smale manifolds, i.e. the $k$-fold connected sum of $S^{2}\times S^{3}.$ Now, we show that LSE metrics exist on Smale manifolds for which $H_{2}(M,\mathbb{Z})_{tor}$ is nontrivial. In particular, we show that most simply-connected positive Sasakian rational homology 5-spheres are also negative Sasakian (hence Lorentzian Sasaki-Einstein). Moreover, we show that for each pair of positive in…
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.
For a conformally compact Poincaré-Einstein manifold $(X,g_+)$, we consider two types of compactifications for it. One is $\bar{g}=ρ^2g_+$, where $ρ$ is a fixed smooth defining function; the other is the adapted (including Fefferman-Graham) compactification $\bar{g}_s=ρ^2_sg_+$ with a continuous parameter $s>\frac{n}{2}$. In this paper, we mainly prove that for a set of conformally compact Poincaré-Einstein manifolds $\{(X, g_{+}^{(i)})\}$ with conformal infinity of positive Yamabe type, $\{\bar{g}^{(i)}\}$ is compact in $C^{k,α}(\overline{X})$ topology if and only if $\{\bar{g}_s^{(i)}\}$ is …
Toric anti-self-dual Einstein metrics via complex geometry
利用扭量对应，我们给出了复曲面反自对偶爱因斯坦度量的一个分类：每一个这样的度量本质上是由一个奇全纯函数决定的。这解释了爱因斯坦度量如何适合于在早期的论文（math.DG/0602423）中给出的一般复曲面反自对偶度量的分类。这些结果补充了Calderbank-Pedersen（数学DG/0105263）的工作，他描述了爱因斯坦度量出现在乔伊斯空间中的位置，导致了不同的分类。利用扭量变换给出了定理的一个新的证明。
Note on the paper of Fu and Wong on strictly pseudoconvex domains with Kähler--Einstein Bergman metrics
证明了Ramadanov猜想蕴涵Cheng猜想。特别是它遵循程猜想举行的二维。
Eta-Einstein condition on twistor spaces of odd-dimensional Riemannian manifolds
本文给出了奇维黎曼流形的扭量空间是eta-Einstein的条件。
This note is devoted to study the implications of nonpositive isotropic curvature and negative Ricci curvature for Einstein $4-$Manifolds.
In this note we prove that a (anti-)self dual quasi Yamabe soliton with positive sectional curvature is rotationally symmetric. This generalizes a recent result of G. Huang and H. Li in dimension four. Whence, (anti-) self dual gradient Yamabe solitons with positive sectional curvature is rotationally symmetric. We also prove that half conformally flat gradient Yamabe soliton has a special warped product structure provided that the potential function has no critical point.
In this short note, we prove that the space of all admissible piecewise linear metrics parameterized by length square on a triangulated manifolds is a convex cone. We further study Regge's Einstein-Hilbert action and give a much more reasonable definition of discrete Einstein metric than our former version in \cite{G}. Finally, we introduce a discrete Ricci flow for three dimensional triangulated manifolds, which is closely related to the existence of discrete Einstein metrics.
In this short note, three infinite families of neutrally stable homogeneous Einstein metrics are ruled out as candidates for local maxima of the Hilbert action.
In this note we adapt the work of Hall to find quasi-Einstein metrics on sphere bundles over products of Fano Kaehler-Einstein manifolds, as well as bundles where only one end is blown down.
In the paper "Einstein metrics on compact simple Lie groups attached to standard triples", the authors introduced the definition of standard triples and proved that every compact simple Lie group $G$ attached to a standard triple $(G,K,H)$ admits a left-invariant Einstein metric which is not naturally reductive except the standard triple $(\Sp(4),2\Sp(2),4\Sp(1))$. For the triple $(\Sp(4),2\Sp(2),4\Sp(1))$, we find there exists an involution pair of $\sp(4)$ such that $4\sp(1)$ is the fixed point of the pair, and then give the decomposition of $\sp(4)$ as a direct sum of irreducible $\ad(4\sp(…
A brief note on how Einstein's general relativity has influenced the development of modern differential geometry
我们简要地回顾了微分几何发展的几个方面，这些方面可能被认为是受到爱因斯坦广义相对论的影响。我们专注于如何爱因斯坦的追求一个完整的几何化的物质和电磁引起了大量的理论工作都在物理学和数学。在这方面，我们也带来光如何最近的研究理论物理导致了新的结果，在一些分支的现代微分几何。
A note on mean curvature, Maslov class and symplectic area of Lagrangian immersions
本文证明了Kähler-Einstein流形中拉格朗日浸入的中曲率形式、辛面积和Maslov类之间的一个简单关系。一个直接的结果是，在具有正数量曲率的凯勒-爱因斯坦流形中，最小拉格朗日浸入是单调的。
In this note, we derive an ordinary differential equation for outgoing light rays in the spacetime of a spherically symmetric contracting dust cloud. The violation of strong cosmic censorship is equivalent to the existence of a solution blowing up at the centre of the symmetry. As an application, we reprove the generic violation of strong cosmic censorship for a dust cloud first proved by Christodoulou. We also derive a similar equation for a spherically symmetric dust cloud in the case of positive cosmological constant and show the generic violation of strong cosmic censorship in this case.
This note illustrates the Ricci flow method based on the Cao.H.D's paper[1] and Yau.S.T's paper[4], and tries to explain the method in detail, especially in some calculations. Jian Song and Weinkove's note[9] used some other estimates to obtain the result, this paper will explain some of their estimates as well. This note was a seminar lecture note in 2022 summer when the author was giving lectures on the geometry analysis seminar reasearching the Ricci flow method. The part of the imporatnt zero order estimate is going to be added in a few days. The level of the author is limited, if there ar…
In [29], Plebanski reformulated the anti-self-dual Einstein equations with non-zero scalar curvature as a first order PDE for a connection in an SO(3)-bundle over the four-manifold. The aim of this article is to place this differential equation in a new framework, in which it is both elliptic and a stationary point of a parabolic flow. To do this, we exploit a link with definite connections (introduced in [12]) to draw an analogy with instantons and the Yang-Mills flow. This picture leads to a natural conjecture, analogous to one made by Donaldson concerning hyperkähler 4-manifolds [9]. It als…
This note studies the equivalencies among convergences of Ricci-flat Kähler-Einstein metrics on Calabi-Yau manifolds, cohomology classes and potential functions.
Crooked planes were defined by Drumm to bound fundamental polyhedra in Minkowski space for Margulis spacetimes. They were extended by Frances to closed polyhedral surfaces in the conformal compactification of Minkowski space (Einstein space) which we call crooked surfaces. The conformal model of anti-de Sitter space is the interior of the quotient of Einstein space by an involution fixing an Einstein plane. The purpose of this note is to show that the crooked planes defined in anti-de Sitter space recently by Danciger-Guéritaud-Kassel lift to restrictions of crooked surfaces in Einstein space …
New properties are derived of renormalized volume functionals, which arise as coefficients in the asymptotic expansion of the volume of an asymptotically hyperbolic Einstein (AHE) manifold. A formula is given for the renormalized volume of an even-dimensional AHE manifold in terms of an arbitrary totally geodesic compactification. The second variation of renormalized volume functionals under conformal change is identified, and is used to show that Einstein metrics of nonzero scalar curvature are local extrema.
In this note, we prove that any non-collapsing and compact Gromov-Hausdorff limit of Kahler-Einstein manifolds is either smooth or is orbifold outside a subvariety of complex codimension at least 3.
In this note we shall prove that the complete Kähler-Einstein volume form on a bounded strongly pseudoconvex domain with $C^{\infty}$-boundary is the normalized limit of a sequence of Bergman kernels.
The Explicit Construction of Einstein Finsler Metrics with Non-Constant Flag Curvature
利用Hawking Taub-NUT度量，本文给出了一个三参数的非常旗曲率的Einstein Finsler度量族的显式构造.
In these notes we give an exposition of a result of G. Tian, which says that a Fano surfaces admits a Kahler-Einstein metric precisely when the Lie algebra of holomorphic vector fields is reductive.
In this note, stimulated by the existence result of Futaki-Ono-Wang for toric Sasaki-Einstein metrics, we obtain new examples of Sasaki-Einstein metrics on S^1-bundles associated to canonical line bundles of P^1-bundles over Kähler-Einstein Fano manifolds, even though the Futaki's obstruction does not vanish. Here the method of Sakane and Koiso is used, and our examples include non-toric Sasaki-Einstein manifolds.
Given a projective structure on a surface $N$, we show how to canonically construct a neutral signature Einstein metric with non-zero scalar curvature as well as a symplectic form on the total space $M$ of a certain rank $2$ affine bundle $M \to N$. The Einstein metric has anti-self-dual conformal curvature and admits a parallel field of anti-self-dual planes. We show that locally every such metric arises from our construction unless it is conformally flat. The homogeneous Einstein metric corresponding to the flat projective structure on $\mathbb{RP}^2$ is the non-compact real form of the Fubi…
Conformally Einstein anti-self-dual spaces and their generalisations
本文研究格拉斯曼几何中爱因斯坦标度与相容度量的存在性，针对反自对偶共形结构的共形到爱因斯坦问题，改进相关条件并推广至(2n,2)-格拉斯曼结构，还得到了相容度量的代数障碍及次极大对称模型的相关性质。
In this note we prove an existence result for the Einstein conformal constraint equations for metrics with vanishing Yamabe invariant assuming that the TT-tensor is small in $L^2$.
Ding stability and Kähler-Einstein metrics on manifolds with big anticanonical class
该研究针对具有大丰沛反典范类的射影流形提出一致Ding稳定性概念，通过建立奇异度量的Deligne泛函理论与Ding泛函斜率公式，证明Kähler-Einstein度量存在性蕴含该稳定性，推广了Fano情形的相关结果并探讨其与模空间的关联。
Almost Complex and Almost Product Einstein Manifolds from a Variational Principle
它表明，一阶（Palatini）变分原理的一般非线性度量仿射拉格朗日依赖于（对称化）Ricci平方不变量导致一个几乎产品爱因斯坦结构或几乎复杂的反厄米爱因斯坦结构的流形上。证明了复流形上的一个实反厄米度量满足Kähler条件，当且仅当该度量是全纯度量的实部分。给出了反Kähler Einstein流形和几乎积Einstein流形的一个刻画.这样的流形的例子被认为是。
In this short note we study flat metric connections with antisymmetric torsion $T \neq 0$. The result has been originally discovered by Cartan/Schouten in 1926 and we provide a new proof not depending on the classification of symmetric spaces. Any space of that type splits and the irreducible factors are compact simple Lie group or a special connection on $S^7$. The latter case is interesting from the viewpoint of $G_2$-structures and we discuss its type in the sense of the Fernandez-Gray classification. Moreover, we investigate flat metric connections of vectorial type.
Quasi-Einstein metrics from Horava gravity
本文揭示了霍拉瓦引力静态正交解与拟爱因斯坦度量间的直接等价关系，并给出逆命题成立的条件。
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