James D. E. Grant

On Lorentzian causality with continuous metrics

We present a systematic study of causality theory on Lorentzian manifolds with continuous metrics. Examples are given which show that some standard facts in smooth Lorentzian geometry, such as light-cones being hypersurfaces, are wrong when metrics which are merely continuous are considered. We show that the existence of time functions remains true on domains of dependence with continuous metrics, and that C0, 1 differentiability of the metric suffices for many key results of the smooth causality theory.

The wave equation on singular space-times

We prove local unique solvability of the wave equation for a large class of weakly singular, locally bounded space-time metrics in a suitable space of generalised functions.

Hypercomplex integrable systems

In this paper we study hypercomplex manifolds in four dimensions. Rather than using an approach based on differential forms, we develop a dual approach using vector fields. The condition on these vector fields may then be interpreted as Lax equations, exhibiting the integrability properties of such manifolds. A number of different field equations for such hypercomplex manifolds are derived, one of which is in Cauchy-Kovaleskaya form which enables a formal general solution to be given. Various other properties of the field equations and their solutions are studied, such as their symmetry properties and the associated hierarchy of conservation laws.

ON SELF-DUAL GRAVITY

We study the Ashtekar-Jacobson-Smolin equations that characterise four-dimensional complex metrics with self-dual Riemann tensor. We find that we can characterise any self-dual metric by a function that satisfies a nonlinear evolution equation, to which the general solution can be found iteratively. This formal solution depends on two arbitrary functions of three coordinates. We study the symmetry algebra of these equations and find that they admit a generalised W∞ ○+ W∞ algebra. We then find the associated conserved quantities which are found to have vanishing Poisson brackets (up to surface terms). We construct explicitly some families of solutions that depend on two free functions of two coordinates, included in which are the multi-center metrics of Gibbons and Hawking

Multidimensional integrable systems and deformations of Lie algebra homomorphisms

We use deformations of Lie algebra homomorphisms to construct deformations of dispersionless integrable systems arising as symmetry reductions of anti-self-dual Yang-Mills equations with a gauge group Diff(S1).

Volume comparison for hypersurfaces in Lorentzian manifolds and singularity theorems

We develop area and volume comparison theorems for the evolution of spacelike, acausal, causally complete hypersurfaces in Lorentzian manifolds, where one has a lower bound on the Ricci tensor along timelike curves, and an upper bound on the mean curvature of the hypersurface. Using these results, we give a new proof of Hawking’s singularity theorem.

On Galilean connections and the first jet bundle

We see how the first jet bundle of curves into affine space can be realized as a homogeneous space of the Galilean group. Cartan connections with this model are precisely the geometric structure of second-order ordinary differential equations under time-preserving transformations — sometimes called KCC-theory. With certain regularity conditions, we show that any such Cartan connection induces “laboratory” coordinate systems, and the geodesic equations in this coordinates form a system of second-order ordinary differential equations. We then show the converse — the “fundamental theorem” — that given such a coordinate system, and a system of second order ordinary differential equations, there exists regular Cartan connections yielding these, and such connections are completely determined by their torsion.

Reducible Connections and Non-local Symmetries of the Self-dual Yang–Mills Equations

We construct the most general reducible connection that satisfies the self-dual Yang–Mills equations on a simply-connected, open subset of flat

Value distribution and spectral theory of Schrödinger operators with L 2 -sparse potentials

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