Einstein structures: Existence versus uniqueness

A Weyl manifold is a conformal manifold equipped with a torsion free connection preserving the conformal structure, called a Weyl connection. It is said to be Einstein-Weyl if the symmetric tracefree part of the Ricci tensor of this connection vanishes. In particular, if the connection is the Levi-Civita connection of a compatible Riemannian metric, then this metric is Einstein. Such an approach has two immediate advantages: firstly, the homothety invariance of the Einstein condition is made explicit by focusing on the connection rather than the metric; and secondly, not every Weyl connection is a Levi-Civita connection, and so Einstein-Weyl manifolds provide a natural generalisation of Einstein geometry. The simplest examples of this generalisation are the locally conformally Einstein manifolds. A Weyl connection on a conformal manifold is said to be closed if it is locally the Levi-Civita connection of a compatible metric; but it need not be a global metric connection unless the manifold is simply connected. Closed EinsteinWeyl structures are then locally (but not necessarily globally) Einstein, and provide an interpretation of the Einstein condition which is perhaps more appropriate for multiply connected manifolds. For example, S1×Sn−1 admits flat Weyl structures, which are therefore closed Einstein-Weyl. These closed structures arise naturally in complex and quaternionic geometry. Einstein-Weyl geometry not only provides a different way of viewing Einstein manifolds, but also a broader setting in which to look for and study them. For instance, few compact Einstein manifolds with positive scalar curvature and continuous isometries are known to have Einstein deformations, yet we shall see that it is precisely under these two conditions that nontrivial Einstein-Weyl deformations can be shown to exist, at least infinitesimally. The Einstein-Weyl condition is particularly interesting in three dimensions, where the only Einstein manifolds are the spaces of constant curvature. In contrast, three dimensional Einstein-Weyl geometry is extremely rich [16, 68, 72], and has an equivalent formulation in twistor theory [34] which provides a tool for constructing selfdual four dimensional geometries. In section 10, we shall discuss a construction relating Einstein-Weyl 3-manifolds and hyperKahler 4-manifolds [40, 29, 50, 79]. Twistor methods also yield complete selfdual Einstein metrics of negative scalar curvature with prescribed conformal infinity [48, 35]. An important special case of this construction is the case of an Einstein-Weyl conformal infinity [34, 61]. Although Einstein-Weyl manifolds can be studied, along with Einstein manifolds, in a Riemannian framework, the natural context is Weyl geometry [23]. We

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.

We study topological obstructions to the existence of Riemannian metrics of non-negative scalar curvature on almost spin manifolds using the Dirac operator, the Bochner technique, C* algebras and von Neumann algebras. We also derive some obstructions in terms of the eta invariants of Atiyah, Patodi and Singer. Next, we prove vanishing theorems for the Atiyah-Milnor genus. Finally, we derive obstructions to the existence of metrics of non-negative scalar curvature along the leaves of a leafwise non-amenable foliation on a spin manifold.

The purpose of this paper is to construct complete Riemannian Manifolds with negative Ricci curvature metric. We shall mostly concentrate on threedimensional manifolds. In [BJ], Bourguignon raised the problem that the connected sum of two Riemannian Manifolds with negative Ricci curvature has a metric with negative Ricci curvature. Yau first observed that the connected sum of two hyperbolic manifolds has a negative Ricei curvature metric. In the first part of the paper, we shall give the affirmative answer to the above question. We actually prove something more. We shall prove that the "connected sum along circles" of two Riemannian Manifolds with negative Ricci curvature metric (Sect. 1) has a negative Ricci curvature metric. In the second part, we only consider three-dimensional manifolds. We use the above results and some topological constructions to obtain new manifolds with negative Ricci curvature metric from a given manifold with negative Ricci curvature metric. Give M a complete Riemannian three-dimensional manifold with negative Ricci curvature metric, then there are complete negative Ricci curvature Riemannian metric on M # S 2 x S I ; M ~ S 2 x S I # L ( p , q ) , and M ~ S 2 x S ~ 1; x S 1, where 2; is any oriented Riemann surface. For the motivation of this paper see [YS]. This paper is an extended version of the second part of the author's Ph.D. thesis at Stony Brook. I would like to thank my advisor Professor Blaine Lawson for suggesting this problem to me, and inspiring discussions and valuable advice.

A flat complex vector bundle (E,D) on a compact Riemannian manifold (X,g) is stable (resp. polystable) in the sense of Corlette [C] if it has no D -invariant subbundle (resp. if it is the D -invariant direct sum of stable subbundles). It has been shown in [C] that the polystability of (E,D) in this sense is equivalent to the existence of a so-called harmonic metric in E . In this paper we consider flat complex vector bundles on compact Hermitian manifolds (X,g) . We propose new notions of g -(poly-)stability of such bundles, and of g -Einstein metrics in them; these notions coincide with (poly-)stability and harmonicity in the sense of Corlette if g is a Kähler metric, but are different in general. Our main result is that the g -polystability in our sense is equivalent to the existence of a g -Hermitian-Einstein metric. Our notion of a g -Einstein metric in a flat bundle is motivated by a correspondence between flat bundles and Higgs bundles over compact surfaces, analogous to the correspondence in the case of Kähler manifolds [S1], [S2], [S3].

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.

Among the known examples of Einstein manifolds, a good many are Kahler. In fact, all compact examples with zero or negative scalar curvature are either Kahler, or locally homogeneous. On a complex manifold, one often gets Kahler-Einstein metrics by specific techniques. One reason is perhaps, in the Kahler case, the relative autonomy of the Ricci tensor with regard to the metric, once the complex structure is given. The Ricci tensor—or, to be precise, the Ricci form—only depends on the volume form. On the other hand, on a compact manifold, the cohomology class of the Ricci form is determined by the complex structure (so that, in particular, the sign of an eventual Kahler-Einstein metric is itself determined by the complex structure). Due to these circumstances, it has been possible to exhibit some existence theorems of Einstein metrics in the Kahler framework (Calabi-Yau and Aubin-Calabi-Yau theorems) which have no counterpart in general Riemannian geometry.

In this paper, we consider a perturbation of the Ricci solitons equation proposed in [J.-P. Bourguignon, Ricci curvature and Einstein metrics, in Global Differential Geometry and Global Analysis, Lecture Notes in Mathematics, Vol. 838 (Springer, Berlin, 1981), pp. 42–63] and studied in [H.-D. Cao, Geometry of Ricci solitons, Chinese Ann. Math. Ser. B27(2) (2006) 121–142] and we classify noncompact gradient shrinkers with bounded non-negative sectional curvature.

Kähler–Einstein currents, also known as singular Kähler–Einstein metrics, have been introduced and constructed a little over a decade ago. These currents live on mildly singular compact Kähler spacesXand their two defining properties are the following: They are genuine Kähler–Einstein metrics on$X_{\mathrm {reg}}$, and they admit local bounded potentials near the singularities ofX. In this note, we show that these currents dominate a Kähler form near the singular locus, when eitherXadmits a global smoothing, or whenXhas isolated smoothable singularities. Our results apply to klt pairs and allow us to show that ifXis any compact Kähler space of dimension three with log terminal singularities, then any singular Kähler–Einstein metric of nonpositive curvature dominates a Kähler form.

We establish the unique complete Kähler–Einstein metric with negative scalar curvature on a broad class of complete Kähler manifolds, including those manifolds whose covering space can be biholomorphically embedded into a Kähler manifold with holomorphic sectional curvature bounded above by a negative constant. We further present several new examples of complete noncompact Kähler–Einstein manifolds, generated by the results.

We prove that any metric of positive scalar curvature on a manifold X extends to the trace of any surgery in codim > 2 on X to a metric of positive scalar curvature which is product near the boundary. This provides a direct way to construct metrics of positive scalar curvature on compact manifolds with boundary. We also show that the set of concordance classes of all metrics with positive scalar curvature on S n is a group.

For a closed manifold M M let ℜ sec + ( M ) \Re _{{\text {sec}}}^ + (M) (resp. ℜ Ric + ( M ) \Re _{{\text {Ric}}}^ + (M) ) be the space of Riemannian metrics on M M with positive sectional (resp. Ricci) curvature and let Diff ( M ) {\text {Diff}}(M) be the diffeomorphism group of M M , which acts on these spaces. We construct examples of 7 7 -dimensional manifolds for which the moduli space ℜ sec + ( M ) / Diff ( M ) \Re _{{\text {sec}}}^ + (M)/{\text {Diff}}(M) is not connected and others for which ℜ Ric + ( M ) / Diff ( M ) \Re _{{\text {Ric}}}^ + (M)/{\text {Diff}}(M) has infinitely many connected components. The examples are obtained by analyzing a family of positive sectional curvature metrics on homogeneous spaces constructed by Aloff and Wallach, on which S U ( 3 ) SU(3) acts transitively, respectively a family of positive Einstein metrics constructed by Wang and Ziller on homogeneous spaces, on which S U ( 3 ) × S U ( 2 ) × U ( 1 ) SU(3) \times SU(2) \times U(1) acts transitively.

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 study the Seiberg-Witten invariant $\\lambda_{\\rm{SW}} (X)$ of smooth spin\n$4$-manifolds $X$ with integral homology of $S^1\\times S^3$ defined by Mrowka,\nRuberman, and Saveliev as a signed count of irreducible monopoles amended by an\nindex-theoretic correction term. We prove a splitting formula for this\ninvariant in terms of the Fr{\\o}yshov invariant $h(X)$ and a certain Lefschetz\nnumber in the reduced monopole Floer homology of Kronheimer and Mrowka. We\napply this formula to obstruct existence of metrics of positive scalar\ncurvature on certain 4-manifolds, and to exhibit new classes of integral\nhomology $3$-spheres of Rohlin invariant one which have infinite order in the\nhomology cobordism group.\n

In the process of finding Einstein metrics in dimension n ≥ 3, we can search critical metrics for the scalar curvature functional in the space of the fixed-volume metrics of constant scalar curvature on a closed oriented manifold. This leads to a system of PDEs (which we call the Fischer–Marsden Equation, after a conjecture concerning this system) for scalar functions, involving the linearization of the scalar curvature. The Fischer–Marsden conjecture said that if the equation admits a solution, the underlying Riemannian manifold is Einstein. Counter-examples are known by O. Kobayashi and J. Lafontaine. However, almost all the counter-examples are homogeneous. Multiple solutions to this system yield Killing vector fields. We show that the dimension of the solution space W can be at most n+1, with equality implying that (M, g) is a sphere with constant sectional curvatures. Moreover, we show that the identity component of the isometry group has a factor SO(W). We also show that geometries admitting Fischer–Marsden solutions are closed under products with Einstein manifolds after a rescaling. Therefore, we obtain a lot of non-homogeneous counter-examples to the Fischer–Marsden conjecture. We then prove that all the homogeneous manifold M with a solution are in this case. Furthermore, we also proved that a related Besse conjecture is true for the compact homogeneous manifolds.