Michael Eastwood

Conformally invariant differential operators on Minkowski space and their curved analogues

This article describes the construction of a natural family of conformally invariant differential operators on a four-dimensional (pseudo-)Riemannian manifold. Included in this family are the usual massless field equations for arbitrary helicity but there are many more besides. The article begins by classifying the invariant operators on flat space. This is a fairly straightforward task in representation theory best solved through the theory of Verma modules. The method generates conformally invariant operators in the curved case by means of Penroses local twistor transport.

Metric Connections in Projective Differential Geometry

We search for Riemannian metrics whose Levi-Civita connection belongs to a given projective class. Following Sinjukov and Mikes, we show that such metrics correspond precisely to suitably positive solutions of a certain projectively invariant finite-type linear system of partial differential equations. Prolonging this system, we may reformulate these equations as defining covariant constant sections of a certain vector bundle with connection. This vector bundle and its connection are derived from the Cartan connection of the underlying projective structure.

PROLONGATIONS OF GEOMETRIC OVERDETERMINED SYSTEMS

We show that a wide class of geometrically defined overdetermined semilinear partial differential equations may be explicitly prolonged to obtain closed systems. As a consequence, in the case of linear equations we extract sharp bounds on the dimension of the solution space.

On Affine Normal Forms and a Classification of Homogeneous Surfaces in Affine Three-Space

We classify homogeneous surfaces in real and complex affine three-space. This is achieved by choosing affine coordinates so that the surface is defined by a function whose Taylor series is in a preferred normal form.

Edge of the Wedge Theory in Hypo-analytic Manifolds

Abstract This article studies microlocal regularity properties of the distributions fon a strongly noncharacteristic submanifold Eof a hypo-analytic manifold Mthat arise as the boundary values of solutions on wedges in Mwith edge E. The hypo-analytic wave-front set of fin the sense of Baouendi-Chang-Treves is constrained as a consequence of the fact that fextends as a solution to a wedge.

Higher Symmetries of the Square of the Laplacian

The symmetry operators for the Laplacian in flat space were recently described and here we consider the same question for the square of the Laplacian. Again, there is a close connection with conformal geometry. There are three main steps in our construction. The first is to show that the symbol of a symmetry is constrained by an overdetermined partial differential equation. The second is to show existence of symmetries with specified symbol (using a simple version of the AdS/CFT correspondence). The third is to compute the composition of two first order symmetry operators and hence determine the structure of the symmetry algebra. There are some interesting differences as compared to the corresponding results for the Laplacian.

Holomorphic realization of ∂-cohomology and constructions of representations☆

Abstract A language is developed for ∂-cohomology, which is different from both the Dolbeault and the Cech descriptions, and involves only holomorphic objects. This language is then illustrated in certain cases of interest to representation theory. This makes possible a new geometric construction of the ladder representations for SU(2, p) and the non-holomorphic discrete series representations of SU(2, 1). The constructions are closely related to Penrose transforms.

COMPLEX ANALYSIS AND THE FUNK TRANSFORM

The Funk transform is defined by integrating a func- tion on the two-sphere over its great circles. We use complex anal- ysis to invert this transform.

Fattening complex manifolds: Curvature and Kodaira—Spencer maps

Abstract We present a calculus whereby the curvature of a geometry arising from any generalized twistor correspondence is related to an obstruction-theoretic classification of the infinitesimal neighborhoods of submanifolds of its twistor space. The crux of the argument involves a relation between Kodaira—Spencer maps and the Penrose transform.

Radon and Fourier transforms for D-modules

The Fourier and Radon hyperplane transforms are closely related, and one such relation was established by Brylinski [4] in the framework of holonomic D-modules. The integral kernel of the Radon hyperplane transform is associated with the hypersurface SCP P of pairs ðx; yÞ; where x is a point in the n-dimensional complex projective space P belonging to the hyperplane yAP : As it turns out, a useful variant is obtained by considering the integral transform associated with the open complement U of S in P P : In the first part of this paper, we generalize Brylinski’s result in order to encompass this variant of the Radon transform, and also to treat arbitrary quasi-coherent D-modules, as well as (twisted) abelian sheaves. Our proof is entirely geometrical, and consists in a reduction to the onedimensional case by the use of homogeneous coordinates. The second part of this paper applies the above result to the quantization of the Radon transform, in the sense of [7]. First we deal with line bundles. More precisely, let P 1⁄4 PðVÞ be the projective space of lines in the vector spaceV; denote by ð Þ 3 D R