Paul Schwahn
We prove the linear stability with respect to the Einstein-Hilbert action of the symmetric spaces $\mathrm{SU}(n)$, $n\geq3$, and $E_6/F_4$. Combined with earlier results, this resolves the stability problem for irreducible symmetric spaces of compact type.
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In this paper, we classify three-locally-symmetric spaces for a connected, compact and simple Lie group. Furthermore, we give the classification of invariant Einstein metrics on these spaces.
Let (M,g) be an Einstein manifold of dimension n \geq 4 with nonnegative isotropic curvature. We show that (M,g) is locally symmetric.
We obtain a locally symmetric Kaehler Einstein structure on a tube in the nonzero cotangent bundle of a Riemannian manifold of positive constant sectional curvature. The obtained Kaehler Einstein structure cannot have constant holomorphic sectional curvature.
We obtain new examples of non-symmetric Einstein solvmanifolds by combining two techniques. In \cite{T2}, H. Tamaru constructs new {\em attached} solvmanifolds, which are submanifolds of the solvmanifolds corresponding to noncompact symmetric spaces, endowed with a natural metric. Extending this construction, we apply it to {\em associated} solvmanifolds, described in \cite{GK}, obtained by modifying the algebraic structure of the solvable Lie algebras corresponding to noncompact symmetric spaces. Our new examples are Einstein solvmanifolds with nilradicals of high nilpotency, which are geomet…
We obtain a class of locally symmetric Kaehler Einstein structures on the cotangent bundle of a Riemannian manifold of negative sectional curvature. Similar results are obtained in the case of a Riemannian manifold of positive sectional curvature. The obtained class of Kaehler Einstein structures depends on one essential parameter and cannot have constant holomorphic sectional curvature.
We show that a connection with skew-symmetric torsion satisfying the Einstein metricity condition exists on an almost contact metric manifold exactly when it is D-homothetic to a cosymplectic manifold. In dimension five, we get that the existence of a connection with skew torsion satisfying the Einstein metricity condition is equivalent to the existence of a Sasaki-Einstein 5-manifold and vice versa, any Sasaki-Einstein 5-manifold generates a two parametric family of connections with skew torsion satisfying the Einstein metricity condition. Formulas for the curvature and the Ricci tensors of t…
We obtain a class of locally symetric Kaehler Einstein structures on the nonzero cotangent bundle of a Riemannian manifold of positive constant sectional curvature. The obtained class of Kaehler Einstein structures depends on one essential parameter and cannot have constant holomorphic sectional curvature.
Study on Einstein Warped Product space with Quarter Symmetric Connection
该文研究带四分之一对称联络的爱因斯坦扭曲空间，推导了相关曲率张量的基本结论，证明了二阶联络相关结果，并给出了该空间退化为黎曼乘积空间的条件。
We obtain a locally symmetric Kaehler Einstein structure on the cotangent bundle of a Riemannian manifold of negative constant sectional curvature. Similar results are obtained on a tube around zero section in the cotangent bundle, in the case of a Riemannian manifold of positive constant sectional curvature. The obtained Kaehler Einstein structures cannot have constant holomorphic sectional curvature.
In this paper, we study the doubly warped product manifolds with semisymmetric metric connection. We derive the curvatures formulas for doubly warped product manifold with semi-symmetric metric connection in terms of curvatures of components of doubly warped product manifolds. We also prove the necessary and sufficient condition for a doubly warped product manifold to be a warped product manifold. We obtain some results for Einstein doubly warped product manifold and Einstein-like doubly warped product manifold of class A with respect to a semi-symmetric metric connection.
We give a classification of compact conformally Kahler Einstein-Weyl manifolds whose Ricci tensor is hermitian.
In this paper, we establish some compactness results of conformally compact Einstein metrics on $4$-dimensional manifolds. Our results were proved under assumptions on the behavior of some local and non-local conformal invariants, on the compactness of the boundary metrics at the conformal infinity, and on the topology of the manifolds.
The goal of this article is to study compact quasi-Einstein manifolds with boundary. We provide boundary estimates for compact quasi-Einstein manifolds simi\-lar to previous results obtained for static and $V$-static spaces. In addition, we show that compact quasi-Einstein manifolds with connected boundary and satisfying a suitable pinching condition must be isometric, up to scaling, to the standard hemisphere $\mathbb{S}_{+}^{n}.$
In this paper, we show that, for every Hermitian vector bundle over a compact Kaehler Einstein manifold, if the projection is biharmonic, then it is harmonic.
We ask a general question: what are locally homogeneous compact pseudo-Riemannian Einstein manifolds? We show that any standard compact Clifford-Klein form of a simple non-compact Lie group admits at least one Einstein metric. We conjecture that these are basically the only possible locally homogeneous Einstein pseudo-Riemannian compact manifolds using T. Kobayashi's conjecture as a guiding principle.
In this paper, we establish compactness results of some class of conformally compact Einstein 4-manifolds. In the first part of the paper, we improve the earlier results obtained by Chang-Ge. In the second part of the paper, as applications, we derive some compactness results under perturbation conditions when the L^2-norm of the Weyl curvature is small. We also derive the global uniqueness of conformally compact Einstein metrics on the 4-Ball constructed in the earlier work of Graham-Lee.
Bochner's technique in Einstein's non-symmetric geometry
本文提出爱因斯坦非对称几何中波彻技术的框架，定义了爱因斯坦连接，推导了魏茨曼克型分解公式，并得出了波彻和霍奇型拉普拉斯算子空空间的消失结果。
Based on the representation theory and the study on the involutions of compact simple Lie groups, we show that $F_4$ admits non-naturally reductive Einstein metrics.
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.
Rigidity of compact four-dimensional weakly Einstein Ricci Solitons
该研究证明紧致四维弱爱因斯坦里奇孤子必为爱因斯坦，非平凡紧致情形归约为梯度收缩情形，通过微分与曲率恒等式推导得势函数关系，不存在紧致真弱爱因斯坦四维流形具里奇孤子结构，非紧致齐次例凸显紧致性假设的必要性。
We show that if $M$ is an Einstein hypersurface in an irreducible Riemannian symmetric space $\overline{M}$ of rank greater than $1$ (the classification in the rank-one case was previously known), then either $\overline{M}$ is of noncompact type and $M$ is a codimension one Einstein solvmanifold, or $\overline{M}=\mathrm{SU}(3)/\mathrm{SO}(3)$ (respectively, $\overline{M}=\mathrm{SL}(3)/\mathrm{SO}(3)$) and $M$ is foliated by totally geodesic spheres (respectively, hyperbolic planes) of $\overline{M}$, with the space of leaves parametrised by a special Legendrian surface in $S^5$ (respectively…
On a compact Einstein-type manifolds with Riemannian foliations
该研究探讨带有黎叶状结构的紧致爱因斯坦型流形，证明其成为全叶状且具分裂性质的必要条件，并给出紧致几乎里奇孤子与紧致爱因斯坦流形上全叶状黎叶状结构的性质。
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.
In this paper we have investigated some aspects of gradient $ρ$-Einstein Ricci soliton in a complete Riemannian manifold. First, we have proved that the compact gradient $ρ$-Einstein soliton is isometric to the Euclidean sphere by showing that the scalar curvature becomes constant. Second, we have showed that in a non-compact gradient $ρ$-Einstein soliton satisfying some integral condition, the scalar curvature vanishes.
Lorentzian symmetric spaces which are Einstein-Yang-Mills with respect to invariant metric connections
本文分类了具有连通非平凡迷向群的四维单连通不可分解洛伦兹对称空间，并求得了其关于不变度量联络与和乐代数对角度量的 Einstein-Yang-Mills 方程解。
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.
In this article, we study the geometry of compact quasi-Einstein manifolds with boundary. We establish sharp boundary estimates for compact quasi-Einstein manifolds with boundary that improve some previous results. Moreover, we obtain a characterization theorem for such manifolds in terms of the surface gravity of the boundary components, which leads to a new sharp geometric inequality. In addition, we prove a boundary estimate for compact quasi-Einstein manifolds with (possibly disconnected) boundary in terms of the Brown-York mass.
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…
In this work, we investigate the geometry and topology of compact Einstein-type manifolds with nonempty boundary. First, we prove a sharp boundary estimate, as consequence we obtain under certain hypotheses that the Hawking mass is bounded from bellow in terms of area. Then we give a topological classification for its boundary. Finally, we prove a gap result for a compact Einstein-type manifold with boundary.
Throughout the history of Einstein manifolds, differential geometers have shown great interest in finding the relationships between curvature and the topology of Einstein manifolds. In the paper, first, we prove that a compact Einstein manifold $(M,g)$ with Einstein constant $α>0$ is a homo-logical sphere when the minimum of its sectional curvatures $> α/(n+ 2)$; in particular, $(M,g)$ is a spherical space form when the minimum of its sectional curvatures $> α/ n$. Second, we prove two propositions (similar to the above ones) for Tachibana numbers of a compact Einstein manifold $(M,g)$ with $α…
Non-compact Einstein manifolds with unimodular isometry group
本文研究带连通幺模李群真等距作用的非紧负爱因斯坦流形，证明其可等距分解为对称空间与紧负爱因斯坦流形的乘积，证明用到极作用、李论与极值原理。
Hermitian-Einstein Metrics on Parabolic Bundles over compact complex surfaces
本文证明了紧致非Kähler曲面（带简单正规交叉除子）及任意维数带光滑除子的非Kähler流形上抛物丛的Kobayashi-Hitchin对应关系。
We define an invariant for compact spin manifolds $X$ of dimension $4k$ equipped with a metric $h$ of positive Yamabe invariant on its boundary. The vanishing of this invariant is a necessary condition for the conformal class of $h$ to be the conformal infinity of a conformally compact Einstein metric on $X$.
On compactness conformally compact Einstein manifolds and uniqueness of Graham-Lee metrics, III
本文研究了维度 $d\ge 4$ 的符合紧致爱因斯坦度量的紧致性及全局唯一性，并探讨了共形不变量的间隙现象。
In this paper, we study the quasi-Einstein and generalized quasi-Einstein warped products with a semi-symmetric non-metric connection. We give the expressions of the Ricci tensors and scalar curvatures for the bases and fibres. In some cases we give some obstructions to the existence of the quasi-Einstein and generalized quasi-Einstein warped products with a semi-symmetric non-metric connection.
For Einstein manifolds with negative scalar curvature admitting an isometric action of a Lie group G with compact, smooth orbit space, we show the following rigidity result: The nilradical N of G acts polarly, and the N-orbits can be extended to minimal Einstein submanifolds. As an application, we prove the Alekseevskii conjecture: Any homogeneous Einstein manifold with negative scalar curvature is diffeomorphic to a Euclidean space.
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
We discuss a number of topics in the area of conformally compact Einstein metrics, mostly centered around the global existence question of finding such metrics with an arbitrarily prescribed conformal infinity. The paper is partly a survey of this area but also presents new results and a number of open problems.
Einstein hypersurfaces are ``very rare" in rank-one symmetric spaces. Damek-Ricci spaces may be viewed as the closest and the most natural generalisations of noncompact rank-one symmetric spaces. We prove that no Damek-Ricci space admits an Einstein hypersurface.
Compact Einstein-type manifolds with parallel Ricci tensor
该研究推导带边界紧致梯度Einstein型流形的Bochner型恒等式，证明平行Ricci曲率下的刚性结果，并给出势函数梯度范数条件以分类这类结构，拓展了Einstein型流形的刚性与分类研究。
Bismut Einstein metrics on compact complex manifolds
本文通过观察Bismut Ricci形式的(2,0)部分为Chern Laplacian特征向量，证明了非零Einstein常数的Bismut Einstein度量必为Kähler Einstein，而零常数的则为Bismut Ricci平坦。
Resolution of compact Einstein orbifolds in general dimensions
研究高维紧致爱因斯坦流形非塌缩序列的Gromov-Hausdorff极限为紧致爱因斯坦orbifold，证明其存在有限奇点，并给出超几何orbifold的负障碍物，简化了Ozuch四维工作中的技术难点。
The study of left-invariant Einstein metrics on compact Lie groups which are naturally reductive was initiated by J. E. D'Atri and W. Ziller in 1979. In 1996 the second author obtained non-naturally reductive Einstein metrics on the Lie group SU(n) for $n \ge 6$, by using a method of Riemannian submersions. In the present work we prove existence of non-naturally reductive Einstein metrics on the compact simple Lie groups SO(n) ($n \geq 11$), $Sp(n)$ ($n \geq 3$), $E_6$, $E_7$, and $E_8$.
The goal of this paper is to study weakly Einstein critical metrics of the volume functional on a compact manifold $M$ with smooth boundary $\partial M$. Here, we will give the complete classification for an $n$-dimensional, $n=3$ or $4,$ weakly Einstein critical metric of the volume functional with nonnegative scalar curvature. Moreover, in the higher dimensional case ($n\geq5$), we will established a similar result for weakly Einstein critical metric under a suitable constraint on the Weyl tensor.
In this paper, we classify the compact locally homogeneous non-gradient $m$-quasi Einstein 3-manifolds. Along the way, we prove that given a compact quotient of a Lie group of any dimension that is $m$-quasi Einstein, the potential vector field $X$ must be left invariant and Killing. We also classify the nontrivial $m$-quasi Einstein metrics that are a compact quotient of be the product of two Einstein metrics. We also show that $S^1$ is the only compact manifold of any dimension which admits a metric which is nontrivially $m$-quasi Einstein and Einstein.
In this paper, we study the solvmanifolds constructed from any parabolic subalgebras of any semisimple Lie algebras. These solvmanifolds are naturally homogeneous submanifolds of symmetric spaces of noncompact type. We show that the Ricci curvatures of our solvmanifolds coincide with the restrictions of the Ricci curvatures of the ambient symmetric spaces. Consequently, all of our solvmanifolds are Einstein, which provide a large number of new examples of noncompact homogeneous Einstein manifolds. We also show that our solvmanifolds are minimal, but not totally geodesic submanifolds of symmetr…
We show that any compact convex simple lattice polytope is the moment polytope of a Kähler-Einstein orbifold, unique up to orbifold covering and homothety. We extend the Wang-Zhu Theorem \cite{WZ} giving the existence of a Kähler-Ricci soliton on any toric monotone manifold on any compact convex simple labelled polytope satisfying the combinatoric condition corresponding to monotonicity. We obtain that any compact convex simple polytope $P\subset \bR^n$ admits a set of inward normals, unique up to dilatation, such that there exists a symplectic potential satisfying the Guillemin boundary condi…
Einstein solvmanifolds as submanifolds of symmetric spaces
本文结合 Ado 定理与 Nash 嵌入定理，为包含 Einstein 和 Ricci 孤立子可解流形的一类可解李群构造同时为等距嵌入的忠实表示，进而证明所有 Einstein 可解流形均可实现为对称空间的子流形。
Spherically symmetric Einstein-scalar-field equations for slowly particle-like decaying null infinity
研究球对称爱因斯坦-标量场方程，证明小且缓慢类粒子衰减初始数据在零无穷远处存在唯一全局解。
We relate the global log canonical threshold of a variety with torus action to the global log canonical threshold of its quotient. We apply this to certain Fano varieties and use Tian's criterion to prove the existence of Kahler-Einstein metrics on them. In particular, we obtain simple examples of Fano threefolds being Kahler-Einstein but admitting deformations without Kahler-Einstein metric.
Symmetric varieties are normal equivarient open embeddings of symmetric homogeneous spaces, and they are interesting examples of spherical varieties. We prove that all smooth Fano symmetric varieties with Picard number one admit Kähler-Einstein metrics by using a combinatorial criterion for K-stability of Fano spherical varieties obtained by Delcroix. For this purpose, we present their algebraic moment polytopes and compute the barycenter of each moment polytope with respect to the Duistermaat-Heckman measure.
Spherically symmetric Einstein-scalar-field equations with potential for wave-like decaying null infinity
本文证明了带势球对称Einstein-标量场方程在标量势V满足特定条件下，小初始数据及类波衰减类光无穷远经典解的全局存在性与唯一性。
Indefinite Kaehler solutions of the Einstein equations are studied, and it is almost completely determined which compact complex surfaces admit such metrics.
In this paper we prove that, under natural assumptions, the scalar curvature of a Kaehler-Einstein metric on a compactification of C^n is strictly positive.
In this paper we study the topology of conformally compact Einstein 4-manifolds. When the conformal infinity has positive Yamabe invariant and the renormalized volume is also positive we show that the conformally compact Einstein 4-manifold will have at most finite fundamental group. Under the further assumption that the renormalized volume is relatively large, we conclude that the conformally compact Einstein 4-manifold is diffeomorphic to $B^4$ and its conformal infinity is diffeomorphic to $S^3$.
In this paper, we prove an existence result for Kähler-Einstein metrics on $\mathbb Q$-Fano compactifications of Lie groups. As an application, we classify $\mathbb Q$-Fano compactifications of $SO_4(\mathbb C)$ which admit a Kähler-Einstein metric with the same volume as that of a smooth Fano compactification of $SO_4(\mathbb C)$.
For a holomorphic vector bundle over a compact Kähler orbifold, the slope stability of the bundle is shown to be equivalent to the existence of a Hermitian-Einstein metric or to the properness of a certain functional introduced by Donaldson.
In this note, we give a way to classify $\mathbb Q$-Fano compactifications of a semisimple group $G$. We will prove that there are only finitely many such $\mathbb Q$-Fano $G$-compactifications, which admits (singular) Kähler-Einstein metrics. As an application, this improves a former result in arXiv:2001.11320.
Sandwich operators and Einstein deformations of compact symmetric spaces related to Jordan algebras
本研究通过构造紧李代数的三明治算子、结合若尔当代数分析，解决了两类对称空间的爱因斯坦度量二阶形变问题，证明了多数可无穷小形变的不可约紧对称空间的非线性不稳定性。
This article describes some geometric invariants and conformal anomalies for conformally compact Einstein manifolds and their minimal submanifolds which have recently been discovered via the Anti-de Sitter/Conformal Field Theory correspondence.
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