Einstein metrics and smooth structures

Non-naturally reductive Einstein metrics on normal homogeneous Einstein manifolds

It is an important problem in differential geometry to find non-naturally reductive homogeneous Einstein metrics on homogeneous manifolds. In this paper, we consider this problem for some coset spaces of compact simple Lie groups. A new method to construct invariant non-naturally reductive Einstein metrics on normal homogeneous Einstein manifolds is presented. In particular, we show that on the standard homogeneous Einstein manifolds, except for some special cases, there exist plenty of such metrics. A further interesting result of this paper is that on some compact semisimple Lie groups, ther…

On A Class of Finsler Metrics of Einstein-Reversibility

In this paper, we introduce the notion of Einstein-reversibility for Finsler met- rics. We study a class of p-power Finsler metrics determined by a Riemann metric and 1-form which are of Einstein-reversibility. It shows that such a class of Finsler metrics of Einstein-reversibility are always Einstein metrics. In particular, we show that all p-power metrics but Randers metrics, square metrics and 2-dimensional square-root metrics, are always Ricci-flat-parallel. Further, the local structure is almost determined for 2-dimensional square-root metrics which are Einsteinian (equivalently, of isotr…

Smooth metric measure spaces, quasi-Einstein metrics, and tractors

We introduce the tractor formalism from conformal geometry to the study of smooth metric measure spaces. In particular, this gives rise to a correspondence between quasi-Einstein metrics and parallel sections of certain tractor bundles. We use this formulation to give a sharp upper bound on the dimension of the vector space of quasi-Einstein metrics, providing a different perspective on some recent results of He, Petersen and Wylie.

On Einstein Kropina metrics

In this paper, a characteristic condition of Einstein Kropina metrics is given. By the characteristic condition, we prove that a non-Riemannian Kropina metric $F=\frac{α^2}β$ with constant Killing form $β$ on an n-dimensional manifold $M$, $n\geq 2$, is an Einstein metric if and only if $α$ is also an Einstein metric. By using the navigation data $(h,W)$, it is proved that an n-dimensional ($n\geq2$) Kropina metric $F=\frac{α^2}β$ is Einstein if and only if the Riemannian metric $h$ is Einstein and $W$ is a unit Killing vector field with respect to $h$. Moreover, we show that every Einstein Kr…

Einstein metrics with anisotropic boundary behaviour

We construct new examples of complete Einstein metrics on balls. At each point of the boundary at infinity, the metric is asymptotic to a homogeneous Einstein metric on a solvable group, which varies with the point at infinity.

Deformation for coupled Kähler-Einstein metrics

The notion of coupled Kähler-Einstein metrics was introduced recently by Hultgren-WittNyström. In this paper we discuss deformation of a coupled Kähler-Einstein metrics on a Fano manifold. In particular we obtain a necessary and sufficient condition for a coupled Kähler-Einstein metric to be deformed to another coupled Kähler-Einstein metric for a Fano manifold admitting non-trivial holomorphic vector fields. In addition we also discuss deformation for a coupled Käher-Einstein metric on a Fano manifold when the complex structure varies.

Kähler-Einstein metrics along the smooth continuity method

We show that if a Fano manifold $M$ is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then $M$ admits a Kähler-Einstein metric. This is a strengthening of the solution of the Yau-Tian-Donaldson conjecture for Fano manifolds by Chen-Donaldson-Sun, and can be used to obtain new examples of Kähler-Einstein manifolds. We also give analogous results for twisted Kähler-Einstein metrics and Kahler-Ricci solitons.

Stability of Einstein metrics on fiber bundles

We study the linear stability of Einstein metrics of Riemannian submersion type. First, we derive a general instability condition for such Einstein metrics and provide some applications. Then we study instability arising from Riemannian product structures on the base. As an application, we estimate the coindex of the Einstein metrics constructed in \cite{WZ90} and \cite{Wan92}. Finally, we investigate more closely the linear stability of Einstein metrics from circle bundle constructions and obtain a rigidity result for linearly stable Einstein metrics of this type.

Homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds

We show that homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds, using that they admit periodic, integrally minimal foliations by homogeneous hypersurfaces. For the geometric flow induced by the orbit-Einstein condition, we construct a Lyapunov function based on curvature estimates which come from real GIT.

Kaehler-Einstein metrics on orbifolds and Einstein metrics on spheres

A construction of Kaehler-Einstein metrics using Galois coverings, studied by Arezzo-Ghigi-Pirola, is generalized to orbifolds. By applying it to certain orbifold covers of P^n which are trivial set theoretically, one obtains new Einstein metrics on odd-dimensional spheres. The method also gives Kaehler-Einstein metrics on degree 2 Del Pezzo surfaces with A_1 or A_2-singularities.

Quasi-Einstein metrics on hypersurface families

We construct quasi-Einstein metrics on some hypersurface families. The hypersurfaces are circle bundles over the product of Fano, Kähler-Einstein manifolds. The quasi-Einstein metrics are related to various gradient Kähler-Ricci solitons constructed by Dancer and Wang and some Hermitian, non-Kähler, Einstein metrics constructed by Wang and Wang on the same manifolds.

Sasaki-Einstein and paraSasaki-Einstein metrics from (κ,μ)-structures

We prove that any non-Sasakian contact metric (κ,μ)-space admits a canonical η-Einstein Sasakian or η-Einstein paraSasakian metric. An explicit expression for the curvature tensor fields of those metrics is given and we find the values of κand μfor which such metrics are Sasaki-Einstein and paraSasaki-Einstein. Conversely, we prove that, under some natural assumptions, a K-contact or K-paracontact manifold foliated by two mutually orthogonal, totally geodesic Legendre foliations admits a contact metric (κ,μ)-structure. Furthermore, we apply the above results to the geometry of tangent sphere b…

Residue formula for an obstruction to Coupled Kähler-Einstein metrics

We obtain a residue formula for an obstruction to the existence of coupled Kähler-Einstein metrics described by Futaki-Zhang. We apply it to an example studied separately by Futaki and Hultgren which is a toric Fano manifold with reductive automorphism, does not admit a Kähler-Einstein metric but still admits coupled Kähler-Einstein metrics.

紧致复流形上的Bismut Einstein度量

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。

Smooth metric measure spaces and quasi-Einstein metrics

Smooth metric measure spaces have been studied from the two different perspectives of Bakry-Émery and Chang-Gursky-Yang, both of which are closely related to work of Perelman on the Ricci flow. These perspectives include a generalization of the Ricci curvature and the associated quasi-Einstein metrics, which include Einstein metrics, conformally Einstein metrics, gradient Ricci solitons, and static metrics. In this article, we describe a natural perspective on smooth metric measure spaces from the point of view of conformal geometry and show how it unites these earlier perspectives within a un…

Smooth Structures and Einstein Metrics on $CP^2#5,6,7\bar{CP^2}$

We show that each of the topological 4-manifolds $CP^2#k\bar{CP^2}, for $k = 6, 7$ admits a smooth structure which has an Einstein metric of scalar curvature $s > 0$, a smooth structure which has an Einstein metric with $s < 0$ and infinitely many non-diffeomorphic smooth structures which do not admit Einstein metrics. We show that there are infinitely many manifolds homeomorphic non-diffeomorphic to $CP^2#5\bar{CP^2}$ which do not admit an Einstein metric. We also exhibit new higher dimensional examples of manifolds carrying Einstein metrics of both positive and negative scalar curvature. The…

Invariant Einstein metrics on Ledger-Obata spaces

In this paper, we study invariant Einstein metrics on Ledger-Obata spaces $F^m/\operatorname{diag}(F)$. In particular, we classify invariant Einstein metrics on $F^4/\operatorname{diag}(F)$ and estimate the number of invariant Einstein metrics on general Ledger-Obata spaces $F^{m}/\operatorname{diag}(F)$.

李群和齐性空间上的不变爱因斯坦克罗皮纳度量

Invariant Einstein Kropina metrics on Lie groups and homogeneous spaces

本文研究李群与齐性空间上的爱因斯坦克罗皮纳度量，提出构造方法，给出SO(n)上的实例，分类三维实李群，并证明射影空间无此类齐性度量。

Finslerian metrics locally conformally $R$-Einstein

Let $R$ be the $hh$-curvature associated with the Chern connection or the Cartan connection. Adopting the pulled-back tangent bundle approach to the Finslerian Geometry, an intrinsic characterization of $R$-Einstein metrics is given. Finslerian metrics which are locally conformally $R$-Einstein are classified.

几乎接触度量三维流形上的弱爱因斯坦和弱 $\eta$-爱因斯坦结构

Weakly Einstein and weakly $η$-Einstein structures on almost contact metric three-manifolds

本文研究几乎接触度量三维流形上的弱爱因斯坦和弱 $\eta$-爱因斯坦结构，给出 Ricci 算子点态分类，证明满足 $Q\xi=0$ 时度量平坦或弱爱因斯坦非爱因斯坦，并分类弱爱因斯坦的单连通齐次接触度量三维流形。

A note on the uniqueness of quasi-Einstein metrics on $\Bbb{H}^{n}\times \Bbb{R}$

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.

The characterizations on a class of weakly weighted Einstein-Finsler metrics

In this paper, we study the weakly weighted Einstein-Finsler metrics. First, we show that weakly weighted Einstein-Kropina metrics must be of isotropic S-curvature with respect to the Busemann-Hausdorff volume form under a certain condition about the weight constants. Then we characterize weakly weighted Einstein-Kropina metrics completely via their navigation expressions or via $α$ and $β$ respectively.

Existence and deformations of Kahler-Einstein metrics on smoothable Q-Fano varieties

We prove the existence of Kahler-Einstein metrics on Q-Gorenstein smoothable, K-polystable Q-Fano varieties, and we show how these metrics behave, in the Gromov-Hausdorff sense, under Q-Gorenstein smoothings.

爱因斯坦度量与等变调和映射：爱因斯坦检测原理

Einstein Metrics and Equivariant Harmonic Maps: The Einstein Detection Principle

本文针对紧致同质性为1的爱因斯坦度量的局部形变理论，构建结合爱因斯坦边值问题内在重述与等变调和映射辅助理论的分析框架，提出爱因斯坦检测原理，辅助调和映射构建框架且不引入额外自由度。

Non-trivial $m$-quasi-Einstein metrics on simple Lie groups

We call a metric $m$-quasi-Einstein if $Ric_X^m$, which replaces a gradient of a smooth function $f$ by a vector field $X$ in $m$-Bakry-Emery Ricci tensor, is a constant multiple of the metric tensor. It is a generalization of Einstein metrics which contains Ricci solitons. In this paper, we focus on left-invariant metrics on simple Lie groups. First, we prove that $X$ is a left-invariant Killing vector field if the metric on a compact simple Lie group is $m$-quasi-Einstein. Then we show that every compact simple Lie group admits non-trivial $m$-quasi-Einstein metrics except $SU(3)$, $E_8$ and…

凯勒-爱因斯坦度量的爱因斯坦变形

Einstein deformations of Kähler Einstein metrics

该研究探讨了负凯勒-爱因斯坦度量的爱因斯坦变形，通过分析凯勒流形的复几何结构，揭示了变形的二级展开与 $h_1^2$ 和柯德亚-斯梅尔曼括号发散性的关系。

Coindex and rigidity of Einstein metrics on homogeneous Gray manifolds

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.

Non-Einstein relative Yamabe metrics

In this paper, we give a sufficient condition for a positive constant scalar curvature metric on a manifold with boundary to be a relative Yamabe metric, which is a natural relative version of the classical Yamabe metric. We also give examples of non-Einstein relative Yamabe metrics with positive scalar curvature.

On the smooth rigidity of almost-Einstein manifolds with nonnegative isotropic curvature

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.

On the uniqueness of Sasaki-Einstein metrics

Let $S$ be a compact Sasakian manifold which does not admit non-trivial Hamiltonian holomorphic vector fields. If there exists an Einstein-Sasakian metric on $S$, then it is unique.

Twisted Kähler-Einstein metrics

We prove an existence result for twisted Kähler-Einstein metrics, assuming an appropriate twisted K-stability condition. An improvement over earlier results is that certain non-negative twisting forms are allowed.

Kähler-Einstein metrics

We survey the theory of Kähler-Einstein metrics, with particular focus on the circle of ideas surrounding the Yau-Tian-Donaldson conjecture for Fano manifolds.

Third order ODEs and four-dimensional split signature Einstein metrics

We construct a family of split signature Einstein metrics in four dimensions, corresponding to particular classes of third order ODEs considered modulo fiber preserving transformations of variables.

Einstein Metrics, Conformal Curvature, and Anti-Holomorphic Involutions

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.

共形乘积上的爱因斯坦度量

Einstein metrics on conformal products

本文研究两个紧致共形流形共形乘积上的爱因斯坦度量，证明在自然几何假设下此类度量必为扭曲乘积度量。

On Einstein Matsumoto metrics

In this paper, the necessary and sufficient conditions for Matsumoto metrics $F=\frac{α^2}{α-β}$ to be Einstein are given. It is shown that if the length of $β$ with respect to $α$ is constant, then the Matsumoto metric $F$ is an Einstein metric if and only if $α$ is Ricci-flat and $β$ is parallel with respect to $α$. A nontrivial example of Ricci flat Matsumoto metrics is given.

Rigidity of Quasi-Einstein Metrics

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.

Kähler-Einstein metrics and higher alpha-invariants

We give a criterion for the existence of a Kähler-Einstein metric on a Fano manifold $M$ in terms of the higher algebraic alpha-invariants $α_{m,k}(M)$.

Invariant Einstein metrics on three-locally-symmetric spaces

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.

On the Stability of Kähler-Einstein Metrics

Using spin$^c$ structure we prove that Kähler-Einstein metrics with nonpositive scalar curvature are stable (in the direction of changes in conformal structures) as the critical points of the total scalar curvature functional. Moreover if all infinitesimal complex deformation of the complex structure are integrable, then the Kähler-Einstein metric is a local maximal of the Yamabe invariant, and its volume is a local minimum among all metrics with scalar curvature bigger or equal to the scalar curvature of the Kähler-Einstein metric.

Geodesic-Einstein metrics and nonlinear stabilities

In this paper, we introduce notions of nonlinear stabilities for a relative ample line bundle over a holomorphic fibration and define the notion of a geodesic-Einstein metric on this line bundle, which generalize the classical stabilities and Hermitian-Einstein metrics of holomorphic vector bundles. We introduce a Donaldson type functional and show that this functional attains its absolute minimum at geodesic-Einstein metrics, and we also discuss the relations between the existence of geodesic-Einstein metrics and the nonlinear stabilities of the line bundle. As an application, we will prove t…

Non-Riemannian Einstein-Randers metrics on $E_6/A_4$ and $E_6/A_1$

In this paper, we first prove that homogeneous spaces $E_6/A_4$ and $E_6/A_1$ admit Einstein metrics which are $Ad(T\times A_1\times A_4)$-invariant, and then show that they admit Non-Riemannian Einstein-Randers metrics.

Non-naturally reductive Einstein metrics on the compact simple Lie group $F_4$

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.

A Kenmotsu metric as a conformal $η$-Einstein soliton

The object of the present paper is to study some properties of Kenmotsu manifold whose metric is conformal $η$-Einstein soliton. We have studied some certain properties of Kenmotsu manifold admitting conformal $η$-Einstein soliton. We have also constructed a 3-dimensional Kenmotsu manifold satisfying conformal $η$-Einstein soliton.

Lower dimensional $S^1$-invariant Kähler-Einstein metrics via integrable structures

We focus on the classical open problem of the classification of Kähler-Einstein manifolds that can be Kähler immersed into a complex projective space endowed with the Fubini-Study metric. In particular, we will deal with such problem in the special case of Kähler- Einstein metrics admitting symmetries of rotational type. This leads to certain integrable distributions allowing a classification of such metrics.

十维球面上的爱因斯坦度量

Einstein metrics on the Ten-Sphere

本文证明了十维球面上存在三个非圆形、非等距的正标量曲率爱因斯坦度量，突破了此前仅六维和八维球面已知此类度量的限制。

Existence of Weak Conical Kähler-Einstein Metrics Along Smooth Hypersurfaces

The existence of \emph{weak conical Kähler-Einstein} metrics along smooth hypersurfaces with angle between $0$ and $2π$ is obtained by studying a smooth continuity method and a \emph{local Moser's iteration} technique. In the case of negative and zero Ricci curvature, the $C^0$ estimate is unobstructed; while in the case of positive Ricci curvature, the $C^0$ estimate obstructed by the properness of the \emph{twisted K-Energy}. As soon as the $C^0$ estimate is achieved, the local Moser iteration could improve the \emph{rough bound} on the approximations to a \emph{uniform $C^2$ bound}, thus pr…

Multiplier Hermitian-Einstein metrics on Fano manifolds of KSM-type

In this article we focus on multiplier Hermitian-Einstein metrics introduced by Mabuchi which include Kähler-Einstein metrics, Kähler-Ricci solitons and Mabuchi solitons as special cases. We also focus on KSM-manifolds, which are introduced by the first author as toric bundles, to establish a criterion for the existence of multiplier Hermitian-Einstein metrics in terms of KSM-data. An explicit example for a KSM-manifold admitting a family of multiplier Hermitian-Einstein metrics is constructed by using a continuous path connecting a Kähler-Ricci soliton and a Mabuchi soliton.

Quasi-Einstein metrics on sphere bundles

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.

Non-trivial $m$-quasi-Einstein metrics on quadratic Lie groups

We call a metric $m$-quasi-Einstein if $Ric_X^m$ (a modification of the $m$-Bakry-Emery Ricci tensor in terms of a suitable vector field $X$) is a constant multiple of the metric tensor. It is a generalization of Einstein metrics which contains Ricci solitons. In this paper, we focus on left-invariant vector fields and left-invariant Riemannian metrics on quadratic Lie groups. First we prove that any left-invariant vector field $X$ such that the left-invariant Riemannian metric on a quadratic Lie group is $m$-quasi-Einstein is a Killing field. Then we construct infinitely many non-trivial $m$-…

On a Class of Two-Dimensional Einstein Finsler Metrics of Vanishing S-Curvature

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 Hermitian Metrics of Non Negative Sectional Curvature

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.

Global obstructions to conformally Einstein metrics in dimension six

We present a global conformal invariant on closed six-manifolds which obstructs the existence of a conformally Einstein metric. We show that this obstruction is nontrivial and, up to multiplication by a constant, is the unique such invariant. This also gives rise to a (possibly trivial) diffeomorphism invariant which obstructs the existence of an Einstein metric. We also discuss global conformal invariants which obstruct the existence of a conformally Einstein metric on closed six-manifolds with infinite fundamental group.

奇异Kähler-Einstein度量与RCD空间

Singular Kähler-Einstein metrics and RCD spaces

该研究聚焦奇异射影簇上的Kähler-Einstein度量，证明在常数量曲率度量的逼近条件下，其光滑部分的度量完备化为非坍塌RCD空间且与原簇同胚，拓展了奇异几何空间的研究范畴。