Variational equations on mixed Riemannian–Lorentzian metrics

Abstract

A class of elliptic–hyperbolic equations is placed in the context of a geometric variational theory, in which the change of type is viewed as a change in the character of an underlying metric. A fundamental example of a metric which changes in this way is the extended projective disc, which is Riemannian at ordinary points, Lorentzian at ideal points, and singular on the absolute. Harmonic fields on such a metric can be interpreted as the hodograph image of extremal surfaces in Minkowski 3-space. This suggests an approach to generalized Plateau problems in three-dimensional space-time via Hodge theory on the extended projective disc. Analogous variational problems arise on Riemannian–Lorentzian flow metrics in fiber bundles (twisted nonlinear Hodge equations), and on certain Riemannian–Lorentzian manifolds which occur in relativity and quantum cosmology. The examples surveyed come with natural gauge theories and Hodge dualities. This paper is mainly a review, but some technical extensions are proven.