Complete surfaces with ends of non positive curvature

The classical Efimov theorem states that there is no C2C2-smoothly immersed complete surface in R3R3 with negative Gauss curvature uniformly separated from zero. Here we analyze the case when the curvature of the complete surface is less that −c2−c2 in a neighborhood of infinity, and prove the surface is topologically a finitely punctured compact surface, the area is finite, and each puncture looks like cusps extending to infinity, asymptotic to rays.