Spacelike convex surfaces with prescribed curvature in (2+1)-Minkowski space
Résumé
We prove existence and uniqueness of solutions to the Minkowski problem in any domain of dependence D in (2+1)-dimensional Minkowski space, provided D is contained in the future cone over a point. Namely, it is possible to find a smooth convex Cauchy surface with prescribed curvature function on the image of the Gauss map. This is related to solutions of the Monge-Amp ere equation det D^2 u(z) = (1/ψ(z))(1 − |z| 2)^(−2) on the unit disc, with the boundary condition u| ∂D = ϕ, for ψ a smooth positive function and ϕ a bounded lower semicontinuous function. We then prove that a domain of dependence D contains a convex Cauchy surface with principal curvatures bounded from below by a positive constant if and only if the corresponding function ϕ is in the Zygmund class. Moreover in this case the surface of constant curvature K contained in D has bounded principal curvatures, for every K < 0. In this way we get a full classification of isometric immersions of the hyperbolic plane in Minkowski space with bounded shape operator in terms of Zygmund functions of ∂D. Finally, we prove that every domain of dependence as in the hypothesis of the Minkowski problem is foliated by the surfaces of constant curvature K, as K varies in (−∞, 0).
Domaines
Mathématiques [math]
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