On the decay rate of the Gauss curvature for isometric immersions

Abstract

We address the problem of global embedding of a two dimensional Riemannian manifold with negative Gauss curvature into three dimensional Euclidean space. A theorem of Efimov states that if the curvature decays too slowly to zero then global smooth immersion is impossible. On the other hand a theorem of J.-X. Hong shows that if decay is sufficiently rapid (roughly like t −(2+δ) for δ > 0) then global smooth immersion can be accomplished. Here we present recent results on applying the method of compensated compactness to achieve a non-smooth global immersion with rough data and we give an emphasis on the role of decay rate of the Gauss curvature.