Einstein-Weyl Geometry

We prove that if $$M^n(n\ge 4)$$ is a compact Einstein manifold whose normalized scalar curvature and sectional curvature satisfy pinching condition $$R_0>\sigma _{n}K_{\max }$$ , where $$\sigma _n\in (\frac{1}{4},1)$$ is an explicit positive constant depending only on $$n$$ , then $$M$$ must be isometric to a spherical space form. Moreover, we prove that if an $$n(\ge {\!\!4})$$ -dimensional compact Einstein manifold satisfies $$K_{\min }\ge \eta _n R_0,$$ where $$\eta _n\in (\frac{1}{4},1)$$ is an explicit positive constant, then $$M$$ is locally symmetric. It should be emphasized that the pinching constant $$\eta _n$$ is optimal when $$n$$ is even. We then obtain some rigidity theorems for Einstein manifolds under $$(n-2)$$ -th Ricci curvature and normalized scalar curvature pinching conditions. Finally we extend the theorems above to Einstein submanifolds in a Riemannian manifold, and prove that if $$M$$ is an $$n(\ge {\!\!4})$$ -dimensional compact Einstein submanifold in the simply connected space form $$F^{N}(c)$$ with constant curvature $$c\ge 0$$ , and the normalized scalar curvature $$R_0$$ of $$M$$ satisfies $$R_0>\frac{A_n}{A_n+4n-8}(c+H^2),$$ where $$A_n=n^3-5n^2+8n$$ , and $$H$$ is the mean curvature of $$M$$ , then $$M$$ is isometric to a standard $$n$$ -sphere.

We discuss manifolds with positive sectimal curvature on which a group acts isometrically with one dimensional quotient. A number of the known examples have this property, but some potential families for new examples in dimension 7 arise as well. We discuss the geometry of these known examples and the connection that the candidates have with self-dual Einstein metrics. There are very few known examples of manifolds with positive sectional curvature. Apart from the compact rank one symmetric spaces, they exist only in dimensions 24 and below and are all obtained as quotients of a compact Lie group equipped with a biinvariant metric under an isometric group action. They consist of certain homogeneous spaces in dimensions 6,7,12,13, and 24 due to Berger [Be], Wallach [Wa], and Aloff–Wallach [AW], and of biquotients in dimensions 6,7, and 13 due to Eschenburg [E1],[E2],[E3] and Bazaikin [Ba]. When trying to find new examples, it is natural to search among manifolds with large isometry group, a program initiated by K. Grove in the 1990s, see [Wi] for a recent survey. Homogeneous spaces with positive curvature were classified in [Be],[Wa],[BB] in the 1970s. The next natural case to study is therefore manifolds on which a group acts isometrically with one dimensional quotient, so called co-homogeneity one manifolds. L. Verdiani [V1, V2] showed that in even dimensions, positively curved cohomogeneity one manifolds are equivariantly diffeomorphic to an isometric action on a rank one symmetric space. In odd dimensions, K. Grove and the author observed in 1998 that there are infinite families among the known nonsymmetric positively curved manifolds that admit isometric cohomogeneity one actions, and suggested a family of potential 7 dimensional candidates P k . In [GWZ], a classification in odd dimensions was carried out, and another family Q k and an isolated manifold R emerged in dimension 7. It is not yet known whether these manifolds admit a cohomogeneity one metric with positive curvature, although they all admit one with nonnegative curvature as a consequence of the main result in [GZ]. In [GWZ], the authors also discovered an intriguing connection that the manifolds P k and Q k have with a family of self-dual Einstein orbifold metrics constructed by Hitchin [Hi1] on S4. They naturally give rise to 3-Sasakian metrics on P k and Q k , which by definition have lots of positive curvature already. The purpose of this survey is threefold. In Section 2, we study the positively curved cohomogeneity one metrics on known examples with positive curvature, including the explicit functions that define the metric. In Section 3, we describe the classification theorem in [GWZ]. It is remarkable that among 7-manifolds where G = S3 ×S3 acts by cohomogeneity one, one has the known positively curved Eschenburg spaces E p , the Berger space B 7, the Aloff–Wallach space W 7, and the sphere S7, and that the candidates P k , Q k , and R all carry such an action as well. We thus carry out the proof in this most intriguing case where G = S3 ×S3 acts by cohomogeneity one on a compact 7-dimensional simply connected manifold. In Section 4, we describe the relationship to Hitchin’s self-dual Einstein metrics. We also discuss some curvature properties of these Einstein metrics and the metrics they define on P k and Q k . The behavior of these metrics, as well as the known metrics with positive curvature, are illustrated in a series of pictures.

In this note, we prove three rigidity results for Einstein manifolds with bounded covering geometry. (1) An almost flat manifold $(M,g)$ must be flat if it is Einstein, i.e. ${\rm Ric}_g =λg$ for some real number $λ.$ (2) A compact Einstein manifold with a non-vanishing and almost maximal volume entropy is hyperbolic. (3) A compact Einstein manifold admitting a uniform local rewinding almost maximal volume is isometric to a space form.

We consider the positive solution of the following semi-linear elliptic equation on the compact Einstein manifolds $M^{n}$ with positive scalar curvature $R_{0}$ \begin{equation*} \Delta_{0}u-\lambda u+f(u)u^{(n+2)/(n-2)}=0, \end{equation*} where $\Delta_{0}$ is the Laplace-Beltrami operator on $M^{n}$. We prove that for $0<\lambda\leq (n-2)R_{0}/(4(n-1))$ and $f'(u)\leq 0$, and at least one of two inequalities is strict, the only positive solution to the above equation is constant. The method here is intrinsic.

A theorem of E. Lerman and S. Tolman, generalizing a result of T. Delzant, states that compact symplectic toric orbifolds are classified by their moment polytopes, together with a positive integer label attached to each of their facets. In this paper we use this result, and the existence of "global" actionangle coordinates, to give an effective parametrization of all compatible toric complex structures on a compact symplectic toric orbifold, by means of smooth functions on the corresponding moment polytope. This is equivalent to parametrizing all toric Kähler metrics and generalizes an analogous result for toric manifolds. A simple explicit description of interesting families of extremal Kähler metrics, arising from recent work of R. Bryant, is given as an application of the approach in this paper. The fact that in dimension four these metrics are self-dual and conformally Einstein is also discussed. This gives rise in particular to a one parameter family of self-dual Einstein metrics connecting the well known Eguchi-Hanson and Taub-NUT metrics.

We simplify the Hitchin's description of SU (2)-invariant self-dual Einstein metrics, making use of the tau function of a related four-pole Schlesinger system.

The aim of this paper is to prove a sharp inequality for the area of a four dimensional compact Einstein manifold (Σ,gΣ) embedded into a complete five dimensional manifold (M5,g) with positive scalar curvatureRand nonnegative Ricci curvature. Under a suitable choice, we have$area(\Sigma)^{\frac{1}{2}}\inf_{M}R \leq 8\sqrt{6}\pi$. Moreover, occurring equality we deduce that (Σ,gΣ) is isometric to a standard sphere ($\mathbb{S}$4,gcan) and in a neighbourhood of Σ, (M5,g) splits as ((-ϵ, ϵ) ×$\mathbb{S}$4,dt2+gcan) and the Riemannian covering of (M5,g) is isometric to$\Bbb{R}$×$\mathbb{S}$4.

Hypersurfaces of constant $2$-mean curvature in spaces of constant sectional curvature are known to be solutions to a variational problem. We extend this characterization to ambient spaces which are Einstein. We then estimate the $2$-mean curvature of certain hypersurfaces in Einstein manifolds. A consequence of our estimates is a generalization of a result, first proved by Chern, showing that there are no complete graphs in the Euclidean space with positive constant $2$-mean curvature.

In this paper we study a special type of pseudo-Riemannian spaces - quasi-Einstein spaces of constant scalar curvature. These spaces are generalizations of known Einstein spaces. We obtained a linear form of the basic equations of the theory of geodetic mappings for these spaces. The studies are conducted locally in tensor form, without restrictions on the sign and signature of the metric tensor.

We show that the boundary of a projectively compact Einstein manifold of dimension [Formula: see text] can be extended by a line bundle naturally constructed from the projective compactification. This extended boundary is such that its automorphisms can be identified with asymptotic symmetries of the compactification. The construction is motivated by the investigation of a new curved orbit decomposition for a [Formula: see text] dimensional manifold which we prove results in a line bundle over projectively compact order one Einstein manifolds. This article is part of a discussion meeting issue 'At the interface of asymptotics, conformal methods and analysis in general relativity'.

If a smooth compact \(4\)-manifold \(M\) admits a Kahler–Einstein metric \(g\) of positive scalar curvature, Gursky (Ann Math 148:315–337, 1998) showed that its conformal class \([g]\) is an absolute minimizer of the Weyl functional among all conformal classes with positive Yamabe constant. Here we prove that, with the same hypotheses, \([g]\) also minimizes of the Weyl functional on a different open set of conformal classes, most of which have negative Yamabe constant. An analogous minimization result is then proved for Einstein metrics \(g\) which are Hermitian, but not Kahler.

Following the ideas of Hitchin on the twistoral approach to 3-dimensional Einstein-Weyl geometry we construct a series of complex surfaces containing rational curves with self-intersection number 2. These mini twistor spaces are obtained by taking an n-fold covering of a neighbourhood of a (1 , n)- curve in the quadric CP1 x CP1 branched along the curve. We describe the corresponding Einstein-Weyl geometry on the parameter space of curves.

Conformal and generalized concircular mappings of Einstein-Weyl manifolds

14pSD-2 Kerr black holes and compact Einstein manifolds

We discuss the rigidity of Einstein manifolds and generalized quasi-Einstein manifolds. We improve a pinching condition used in a theorem on the rigidity of compact Einstein manifolds. Under an additional condition, we confirm a conjecture on the rigidity