Complete surfaces with negative extrinsic curvature

Abstract

N. V. Efimov \cite{Ef1} proved that there is no complete, smooth surface in ${\mathbb{R}}^3$ with uniformly negative curvature. We extend this to isometric immersions in a 3-manifold with pinched curvature: if $M^3$ has sectional curvature between two constants $K_2$ and $K_3$, then there exists $K_1<\min (K_2,0)$ such that $M$ contains no smooth, complete immersed surface with curvature below $K_1$. Optimal values of $K_1$ are determined. This results rests on a phenomenon of propagations for degenerations of solutions of hyperbolic Monge-Amp{è}re equations.

38 pages, 6 figures