Nonexistence of nonpositively curved surfaces with one embedded end (Q1576584)

A surface in \(\mathbb R^3\) is ``one ended'' if the complement of a sufficiently large compact subset is homeomorphically a punctured disk. The main result of this very well written paper states that a complete, one-ended, nonpositively curved Riemannian surface with only isolated parabolic points, with one end embedded and integrable square length of the fundamental form cannot be \(\mathcal{C}^2\) isometrically immersed in \(\mathbb{R}^3\). Examples are then provided to show that each assumption is indeed necessary and cannot be improved.