  N. V. Efimov \cite{Ef1} proved that there is no complete, smooth surface in
$\R^3$ 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{\`e}re equations.
