We prove existence and uniqueness of solutions to the Minko
wski problem in any domain of dependence
D in
w the MathML source">(2+1)-dimensional Minko
wski 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–Ampère equation
w the MathML source">detD2u(z)=(1/ψ(z))(1−|z|2)−2 on the unit disc,
with the boundary condition
w the MathML source">u|∂D=φ, for
ψ a smooth positive function and
φ a bounded lo
wer 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 w the MathML source">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 w the MathML source">∂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 w the MathML source">(−∞,0).