Charles P. Boyer

Killing vectors in self‐dual, Euclidean Einstein spaces

Using the formalism of complex H‐spaces, we show that all real, Euclidean self‐dual spaces that admit (at least) one Killing vector may be gauged so that only two distinct types of Killing vectors appear; in Kahler coordinates these are the generators of a translational or a rotational symmetry. We give explicit forms both for the Killing vectors and for the constraint on the Kahler potential function Ω which allows for such a Killing vector. In the translational case we show how all such spaces are determined by the general solution of the three‐dimensional, flat Laplace’s equation and how these are related to the multi‐Taub–NUT metrics of Gibbons and Hawking. In the rotational case we simplify the equation determining Ω, but this is not sufficient to obtain the general solution.

ON SASAKIAN–EINSTEIN GEOMETRY

We introduce a multiplication ⋆ (we call it a join) on the space of all compact Sasakian-Einstein orbifolds and show that has the structure of a commutative associative topological monoid. The set of all compact regular Sasakian–Einstein manifolds is then a submonoid. The set of smooth manifolds in is not closed under this multiplication; however, the join of two Sasakian–Einstein manifolds is smooth under some additional conditions which we specify. We use this construction to obtain many old and new examples of Sasakain–Einstein manifolds. In particular, in every odd dimension greater that five we obtain spaces with arbitrary second Betti number.

On Eta-Einstein Sasakian Geometry

A compact quasi-regular Sasakian manifold M is foliated by one-dimensional leaves and the transverse space of this characteristic foliation is necessarily a compact Kähler orbifold . In the case when the transverse space is also Einstein the corresponding Sasakian manifold M is said to be Sasakian η-Einstein. In this article we study η-Einstein geometry as a class of distinguished Riemannian metrics on contact metric manifolds. In particular, we use a previous solution of the Calabi problem in the context of Sasakian geometry to prove the existence of η-Einstein structures on many different compact manifolds, including exotic spheres. We also relate these results to the existence of Einstein-Weyl and Lorenzian Sasakian-Einstein structures.

Canonical Sasakian Metrics

Let M be a closed manifold of Sasaki type. A polarization of M is defined by a Reeb vector field, and for any such polarization, we consider the set of all Sasakian metrics compatible with it. On this space we study the functional given by the square of the L2-norm of the scalar curvature. We prove that its critical points, or canonical representatives of the polarization, are Sasakian metrics that are transversally extremal. We define a Sasaki-Futaki invariant of the polarization, and show that it obstructs the existence of constant scalar curvature representatives. For a fixed CR structure of Sasaki type, we define the Sasaki cone of structures compatible with this underlying CR structure, and prove that the set of polarizations in it that admit a canonical representative is open. We use our results to describe fully the case of the sphere with its standard CR structure, showing that each element of its Sasaki cone can be represented by a canonical metric; we compute their Sasaki-Futaki invariant, and use it to describe the canonical metrics that have constant scalar curvature, and to prove that only the standard polarization can be represented by a Sasaki-Einstein metric.

A note on toric contact geometry

Abstract After observing that the well-known convexity theorems of symplectic geometry also hold for compact contact manifolds with an effective torus action whose Reeb vector field corresponds to an element of the Lie algebra of the torus, we use this fact together with a recent symplectic orbifold version of Delzant’s theorem due to Lerman and Tolman [E. Lerman, S. Tolman, Trans. Am. Math. Soc. 349 (10) (1997) 4201–4230] to show that every such compact toric contact manifold can be obtained by a contact reduction from an odd dimensional sphere.

A Note on Hyperhermitian Four-Manifolds

On montre que les seules 4-varietes hyperhermitiennes sont, a une equivalence conforme pres, des tores et des surfaces K3 avec leurs structures hyperkahler standards et certaines surfaces de Hopf conformement plates

Einstein manifolds and contact geometry

We show that every K-contact Einstein manifold is SasakianEinstein and discuss several corollaries of this result.

Lie theory and separation of variables. 6. The equation iUt + Δ2U = 0

This paper constitutes a detailed study of the nine−parameter symmetry group of the time−dependent free particle Schrodinger equation in two space dimensions. It is shown that this equation separates in exactly 26 coordinate systems and that each system corresponds to an orbit consisting of a commuting pair of first− and second−order symmetry operators. The study yields a unified treatment of the (attractive and repulsive) harmonic oscillator, linear potential and free particle Hamiltonians in a time−dependent formalism. Use of representation theory for the symmetry group permits simple derivations of addition and expansion theorems relating various solutions of the Schrodinger equation, many of which are new.

The topology of instanton moduli spaces. I: The Atiyah-Jones conjecture

In this paper we study the global geometry and topology of the moduli spaces of based SU(2)-instantons over the 4-sphere S4 . These instanton moduli spaces have a rich history and have been analyzed from many points of view. Originally these spaces, which we denote by Mk, were defined as solution spaces (modulo gauge equivalence) to certain partial differential equations, namely the self-duality equations associated to the Yang-Mills functional in SU(2) gauge theory. They have been successfully studied from this point of view by Taubes ([T1], [T2]), Uhlenbeck [U] and others. An important alternative approach was initiated by Ward [W], who related instantons to certain holomorphic bundles on CP3, and was continued by Atiyah and Ward in [AW]. This allowed the classification of instantons on

An infinite hierarchy of conservation laws and nonlinear superposition principles for self‐dual Einstein spaces

4 in terms of quaternionic linear algebra by Atiyah, Drinfeld, Hitchin and Manin [ADHM]. This holomorphic approach was further extended by Donaldson [D], who showed that these bundles were determined by their restriction to a Cp2 and that the restricted bundles only had to satisfy the constraint of being trivial on a fixed line in C2R2. Hurtubise [Hul] then exploited this fact to study the moduli spaces Mk, as did Atiyah [A] to show that Mk arise naturally in the theory of holomorphic maps into loop groups; this latter approach was continued by Gravesen [G]. Atiyah and Jones [AJ] obtained the first results and formulated the foundational questions on the global topology of these moduli spaces. Recall that an element of Mk is a based gauge-equivalence class of a connection on the principal SU(2) bundle over