  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 3-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 singular 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.
