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 (mathbb {R}^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 |K|=+infty ) and (Kle -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.