Université Paris-Est Université Paris-Est - Marne-la-Vallée Université Paris-Est - Créteil Val-de-Marne Centre National de la Recherche Scientifique

Constant curvature surfaces in $(2+1)$-Minkowski space

06/02/2017 - 13:30 - 14:30
SEPPI Andrea
Université de Pavie

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.