Harmonic fields on mixed Riemannian-Lorentzian manifolds
نویسنده
چکیده
The extended projective disc is Riemannian at ordinary points, Lorentzian at ideal points, and singular on the absolute. Harmonic fields on this 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. Boundary-value problems for harmonic fields tend to be over-determined on the Lorentzian part of the domain. Nevertheless, it is possible to show the existence of solutions to such problems in a suitable sense. We give a variety of sufficient conditions for solutions, and indicate extensions to singular Riemannian-Lorentzian manifolds which arise in relativity and quantum cosmology. Sections 1-5 of the paper review aspects of variational theory for elliptic-hyperbolic equations on manifolds; in Sec. 6, new existence and non-existence theorems are proven. MSC2000 : 35M10, 53A10 ∗email: [email protected]
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Variational Equations on Mixed Riemannian-lorentzian Metrics
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