Résume | We will discuss the problem of existence and uniqueness of surfaces
of negative constant (or prescribed) Gaussian curvature K in
(2+1)-dimensional Minkowski space. The simplest example, for K=-1, is
the well-known embedding of hyperbolic plane as the one-sheeted
hyperboloid; however, as a striking difference with the sphere in
Euclidean space, in Minkowski space there are many non-equivalent
isometric embeddings of the hyperbolic plane.
This problem is related to solutions of the Monge-Ampère equation
$\det D^2 u(z)=(1/|K|)(1-|z|^2)^*-2*$ on the unit disc. We will prove
the existence of surfaces with the condition u=f on the boundary of
the disc, for f a bounded lower semicontinuous function. If the
curvature K=K(z) depends smoothly on the point z, this gives a
solution to the so-called Minkowski problem.
On the other hand, we will prove that, for K constant, the principal
curvatures of a K-surface are bounded from below by a positive
constant if and only if the corresponding function f is in the Zygmund
class. Time permitting, we will discuss some generalizations to
constant affine curvature. |