# Séminaires : Séminaire de Géométrie

 Equipe(s) : gd, Responsables : L. Hauswirth, P. Laurain, R. Souam, E. Toubiana Email des responsables : Salle : https://bbb-front.math.univ-paris-diderot.fr/recherche/pau-6ha-of4-mea Adresse : Sophie Germain Description Archive avant 2014 Hébergé par le projet Géométrie et Dynamique de l’IMJ-PRG

 Orateur(s) Andrea SEPPI - University of Pavia, Titre Constant curvature surfaces in (2+1)-Minkowski space Date 06/02/2017 Horaire 13:30 à 15:00 Diffusion 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. Salle https://bbb-front.math.univ-paris-diderot.fr/recherche/pau-6ha-of4-mea Adresse Sophie Germain