Complete negatively curved immersed ends in ${\mathbb{R}}^3$

This paper extends, in a sharp way, the famous Efimov’s Theorem to immersed ends in ${\mathrm{\Re }}^3$. More precisely, let $M$ be a non-compact connected surface with compact boundary. Then there is no complete isometric immersion of $M$ into ${\mathbb{R}}^3$ satisfying that ${\int }_M\mathrm{\mid }K\mathrm{\mid }=+\mathrm{\infty }$ and $K\le -\kappa <0$, where $\kappa$ is a positive constant and $K$ is the Gaussian curvature of $M$. In particular Efimov’s Theorem holds for complete Hadamard immersed surfaces, whose Gaussian curvature $K$ is bounded away from zero outside a compact set.