McKenzie Y. Wang

The initial value problem for cohomogeneity one Einstein metrics

The PDE Ric(g) = λ · g for a Riemannian Einstein metric g on a smooth manifold M becomes an ODE if we require g to be invariant under a Lie group G acting properly on M with principal orbits of codimension one. A singular orbit of the G-action gives a singularity of this ODE. Generically, an equation with such type of singularity has no smooth solution at the singularity. However, in our case, the very geometric nature of the equation makes it solvable. More precisely, we obtain a smooth G-invariant Einstein metric (with any Einstein constant λ) in a tubular neighbourhood around a singular orbit Q ⊂ M for any prescribed G-invariant metric gQ and second fundamental form LQ on Q, provided that the following technical condition is satisfied (which is very often the case): the representations of the principal isotropy group on the tangent and the normal space of the singular orbit Q have no common sub-representations. This Einstein metric is not uniquely determined by the initial data gQ and LQ; in fact, one may prescribe initial derivatives of higher degree, and examples show that this degree can be arbitrarily high. The proof involves a blend of ODE techniques and representation theory of the principal and singular isotropy groups.

On Non-Simply Connected Manifolds with Non-Trivial Parallel Spinors

We examine the possibilities of the full holonomy groups of locally irreducible but not necessarily complete Riemannian spin manifolds admitting a non-trivial parallel spinor and discuss some applications of this classification.

Symmetric spaces and strongly isotropy irreducible spaces

A strongly isotropy irreducible homogeneous space is a connected effective homogeneous space L/H where H is a compact subgroup of a Lie group L such that the identity component H 0 of H acts irreducibly on T~H(L/H). If we assume instead that H, but not necessarily H 0, acts irreducibly on Tetc(L/H), then L/H is called an isotropy irreducible homogeneous space. These spaces are interesting because by Schurs Lemma they all admit an L-invariant Einstein metric, and because they include the irreducible Riemannian symmetric spaces as a sub-family. The nonsymmetric strongly isotropy irreducible homogeneous spaces were classified independently by Manturov [M1,2,3], Wolf [Wol], and later by Kr~mer [Kr], with some minor omissions in the first two references. In addition, in [Wol] a detailed study of the geometric properties of strongly isotropy irreducible spaces was carried out. The isotropy irreducible spaces which are not strongly isotropy irreducible have recently been classified in [WZ3]. In a different direction, we present in this paper a direct proof of a beautiful observation of Wall concerning a correspondence between the compact simply connected irreducible symmetric spaces on the one hand and the compact strongly isotropy irreducible quotients of the classical groups on the other hand (cf. the remarks added in proof in [Wol, pp. 147-148]). This correspondence has a number of exceptions: certain Grassmannians and the isotropy irreducible space SO(7)/G 2, which is diffeomorphic to ~pT. These exceptions appear at first sight to be even more mysterious than the correspondence. But our main result is that there is an explanation, using general principles, of this correspondence as well as all its exceptions. In particular, this means that the classification of the non-symmetric strongly isotropy irreducible quotients of the classical groups may be read off from that of the irreducible symmetric spaces. Consider a compact simply connected irreducible symmetric space G/K, where G is the identity component of the isometry group, and K is the isotropy group,

The cohomogeneity one Einstein equations and Painlevé analysis

Abstract We apply techniques of Painleve–Kowalewski analysis to a Hamiltonian system arising from symmetry reduction of the Ricci-flat Einstein equations. In the case of doubly warped product metrics on a product of two Einstein manifolds over an interval, we show that the cases when the total dimension is 10 or 11 are singled out by our analysis.

Non-Kähler expanding Ricci solitons, Einstein metrics, and exotic cone structures

We produce new non-Kähler, non-Einstein, complete expanding gradient Ricci solitons with conical asymptotics and underlying manifold of the form R × M2 × · · · × Mr, where r ≥ 2 and Mi are arbitrary closed Einstein spaces with positive scalar curvature. We also find numerical evidence for complete expanding solitons on the vector bundles whose sphere bundles are the twistor or Sp(1) bundles over quaternionic projective space. Mathematics Subject Classification (2000): 53C25, 53C44 0. Introduction In [BDGW] we constructed complete steady gradient Ricci soliton structures (including Ricci-flat metrics) on manifolds of the form R ×M2 × . . . × Mr where Mi, 2 ≤ i ≤ r, are arbitrary closed Einstein manifolds with positive scalar curvature. We also produced numerical solutions of the steady gradient Ricci soliton equation on certain non-trivial R and R bundles over quaternionic projective spaces. In the current paper we will present the analogous results for the case of expanding solitons on the same underlying manifolds. Recall that a gradient Ricci soliton is a manifold M together with a smooth Riemannian metric g and a smooth function u, called the soliton potential, which give a solution to the equation: (0.1) Ric(g) + Hess(u) + ǫ 2 g = 0 for some constant ǫ. The soliton is then called expanding, steady, or shrinking according to whether ǫ is greater, equal, or less than zero. A gradient Ricci soliton is called complete if the metric g is complete. The completeness of the vector field ∇u follows from that of the metric (cf [Zh]). If the metric of a gradient Ricci soliton is Einstein, then either Hessu = 0 (i.e., ∇u is parallel) or we are in the case of the Gaussian soliton (cf [PW] or [PRS]). At present most examples of non-Kählerian expanding solitons arise from left-invariant metrics on nilpotent and solvable Lie groups (resp. nilsolitons, solvsolitons), as a result of work by J. Lauret [La1], [La3], M. Jablonski [Ja], and many others (cf the survey [La2]). These expanders are however not of gradient type, i.e., they satisfy the more general equation (0.2) Ric(g) + 1 2 LXg + ǫ 2 g = 0 where X is a vector field on M and L denotes Lie differentiation. A large class of complete, non-Einstein, non-Kählerian expanders of gradient type (with dimension ≥ 3) consists of an r-parameter family of solutions to (0.1) on R ×M2 × . . . ×Mr where k > 1 and Mi are positive Einstein manifolds. The special case r = 1 (i.e., no Mi) is due to Bryant [Bry] and the solitons have positive sectional curvature. The r = 2 case is due to Gastel and Kronz [GK], who adapted Böhm’s construction of complete Einstein metrics with negative scalar curvature to the soliton case. The case of arbitrary r was treated in [DW3] via a generalization Date: revised November 21, 2013. M. Wang is partially supported by NSERC Grant No. OPG0009421. 1 2 M. BUZANO, A. S. DANCER, M. GALLAUGHER, AND M. WANG of the dynamical system studied by Bryant. The soliton metrics in this family are all of multiple warped product type. In other words, the manifold is thought of as being foliated by hypersurfaces of the form Sk ×M2× . . .×Mr each equipped with a product metric depending smoothly on a real parameter t. More recently, Schulze and Simon [SS] constructed expanding gradient Ricci solitons with nonnegative curvature operator in arbitrary dimensions by studying the scaling limits of the Ricci flow on complete open Riemannian manifolds with non-negative bounded curvature operator and positive asymptotic volume ratio. As pointed out in [BDGW], the situation of multiple warped products on nonnegative Einstein manifolds is rather special because of the automatic lower bound on the scalar curvature of the hypersurfaces. This leads, in the case where all factors have positive scalar curvature, i.e., k > 1, to definiteness of certain energy functionals occurring in the analysis of the dynamical system arising from (0.1), and hence to coercive estimates on the flow. In the present case, where one factor is a circle, i.e., k = 1, we can pass, as in [BDGW], to a subsystem where coercivity holds, and this is enough for the analysis to proceed. The new solitons obtained, like those of [DW3], have conical asymptotics, and are not of Kähler type (Theorem 2.14). We note that the lowest dimensional solitons we obtain form a 2-parameter family on R × S2. The special case r = 1 was analysed earlier by the physicists Gutperle, Headrick, Minwalla and Schomerus [GHMS]. As in [BDGW] we also obtain a family of solutions to our soliton equations that yield complete Einstein metrics of negative scalar curvature (Theorem 3.1). These are analogous to the metrics discovered by Böhm in [Bo]. Recall that for Böhm’s construction the fact that the hyperbolic cone over the product Einstein metric on the hypersurface acts as an attractor plays an important role in the convergence proof for the Einstein trajectories. When k = 1, however, no product metric on the hypersurface can be Einstein with positive scalar curvature, so the hyperbolic cone construction cannot be exploited directly. It turns out that the analysis of the soliton case already contains most of the analysis required for the Einstein case. The new Einstein metrics we obtain have exponential volume growth. Since the underlying smooth manifolds in the present paper are identical to those in [BDGW], our constructions give rise to pairs of homeomorphic but not diffeomorphic non-Einstein expanding gradient Ricci solitons as well as similar pairs of complete Einstein manifolds with negative scalar curvature. Furthermore, since our expanders and Einstein metrics have asymptotically conical structures, we also obtain pairs whose asymptotic cones are homeomorphic but not diffeomorphic. The details can be found at the end of §3. To make further progress in the search for expanders, we need to consider more complicated hypersurface types where the scalar curvature may not be bounded below. In [BDGW] we carried out numerical investigations of steady solitons where the hypersurfaces are the total spaces of Riemannian submersions for which the hypersurface metric involves two functions, one scaling the base and one the fibre of the submersion. We now look numerically at expanding solitons with such hypersurface types, in particular where the hypersurfaces are S2 or S3 bundles over quaternionic projective space. We produce numerical evidence of complete expanding gradient Ricci soliton structures in these cases. Before undertaking our theoretical and numerical investigations, we first prove some general results about expanding solitons of cohomogeneity one type. Some of the results follow from properties of general expanding gradient Ricci solitons. However, the proofs are much simpler and sometimes the statements are sharper, which is helpful in numerical studies. The results include monotonicity and concavity properties for the soliton potential similar to those proved in [BDGW] in the steady case, as well as an upper bound for the mean curvature of the hypersurfaces. To derive this bound, we need to know that complete non-Einstein expanding gradient Ricci solitons have infinite volume. We include a proof of this fact here (Prop. 1.22) since we were not able to NON-KÄHLER EXPANDING RICCI SOLITONS II 3 find an explicit statement in the literature. Finally we derive an asymptotic lower bound for the gradient of the soliton potential, which is in turn used to exhibit a general Lyapunov function for the cohomogeneity one expander equations. 1. Background on cohomogeneity one expanding solitons We briefly review the formalism [DW1] for Ricci solitons of cohomogeneity one. We work on a manifold M with an open dense set foliated by equidistant diffeomorphic hypersurfaces Pt of real dimension n. The dimension of M , the manifold where we construct the soliton, is therefore n+1. The metric is then of the form ḡ = dt2 + gt where gt is a metric on Pt and t is the arclength coordinate along a geodesic orthogonal to the hypersurfaces. This set-up is more general than the cohomogeneity one ansatz, as it allows us to consider metrics with no symmetry provided that appropriate additional conditions on Pt are satisfied, see the following as well as Remarks 2.18 and 3.18 in [DW1]. We will also suppose that u is a function of t only. We let rt denote the Ricci endomorphism of gt, defined by Ric(gt)(X,Y ) = gt(rt(X), Y ) and viewed as an endomorphism via gt. Also let Lt be the shape operator of the hypersurfaces, defined by the equation ġt = 2gtLt where gt is regarded as an endomorphism with respect to a fixed background metric Q. The Levi-Civita connections of ḡ and gt will be denoted by ∇ and ∇ respectively. The relative volume v(t) is defined by dμgt = v(t) dμQ We assume that the scalar curvature St = tr(rt) and the mean curvature tr(Lt) (with respect to the normal ν = ∂ ∂t) are constant on each hypersurface. These assumptions hold, for example, if M is of cohomogeneity one with respect to an isometric Lie group action. They are satisfied also when M is a multiple warped product over an interval. The gradient Ricci soliton equation now becomes the system − tr(L̇)− tr(L) + ü+ ǫ 2 = 0, (1.1) r − (trL)L− L̇+ u̇L+ ǫ 2 I = 0, (1.2) d(trL) + δ∇L = 0. (1.3) The first two equations represent the components of the equation in the directions normal and tangent to the hypersurfaces P , respectively. The third equation represents the equation in mixed directions—here δ∇L denotes the codifferential for TP -valued 1-forms. In the warped product case the final equation involving the codifferential automatically holds. This is also true for cohomogeneity one metrics that are monotypic, i.e., when there are no repeated real irreducible summands in the isotropy representation of the principal orbits (cf [BB], Prop. 3.18). There is a conservation law (1.4) ü+ (−u̇+ trL) u̇− ǫu = C for some constant C. Using our equations we may rewrite this as (1.5) S + tr(L)− (u̇− trL) − ǫu+ 1

Classification of Superpotentials

We extend our previous classification [DW4] of superpotentials of “scalar curvature type” for the cohomogeneity one Ricci-flat equations. We now consider the case not covered in [DW4], i.e., when some weight vector of the superpotential lies outside (a scaled translate of) the convex hull of the weight vectors associated with the scalar curvature function of the principal orbit. In this situation we show that either the isotropy representation has at most 3 irreducible summands or the first order subsystem associated to the superpotential is of the same form as the Calabi-Yau condition for submersion type metrics on complex line bundles over a Fano Kähler-Einstein product.

On conserved quantities of certain cohomogeneity one Ricci-flat equations

We give a classification of continuously differentiable on-shell integrals which are linear or quadratic in momenta for a Hamiltonian system with constraint associated to the Ricci-flat cohomogeneity one Einstein equations for two special classes of principal orbits/hypersurfaces. Relations to known on-shell integrals and to Painlevé analysis are discussed.

Painlevé expansions and the Einstein equations: the two-summand case

Abstract We investigate the existence of Painleve–Kovalevskaya expansions for various reductions to ordinary differential equations of the Ricci-flat equations. We investigate links between such expansions and metrics of exceptional holonomy.

Ricci-flat warped products and Painlevé analysis

We apply Painleve analysis to the Ricci-flat Einstein equations for a warped product with an arbitrary number of factors. We find that, as in the situation of the two factors examined [J. Geom. Phys. 38, 183–206 (2001)], the cases when the total dimension is 10 or 11 are singled out by the analysis.

Cohomogeneity one Ricci solitons

We introduce a general formalism for studying cohomogeneity on Ricci solitons, and illustrate this by producing new families of examples.