Spacelike convex surfaces with prescribed curvature in (2+1)-Minkowski space

作者：Bonsante Francesco^*; Seppi Andrea

来源：Advances in Mathematics, 2017, 304: 434-493.

DOI：10.1016/j.aim.2016.09.003

• Monge-Ampere equation
• Constant curvature surfaces
• Minkowski space
• Universal Teichmuller theory

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 Ampere equation det D(2)u(z). = (1/psi(z))(1 - vertical bar z vertical bar(2))(-2) on the unit disc, with the boundary condition u vertical bar(partial derivative D) = phi, for psi a smooth positive function and phi a bouncled lower seinicoutirmous 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 phi 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 partial derivative 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 (-infinity, 0).

• 出版日期2017-1-2