Surfaces of constant curvature in 3-dimensional space forms

Abstract

This paper is a survey of some classical results on complete surfaces with constant Gaussian curvature in 3-dimensional space forms. Specifically, in Section 3 the author deals with complete surfaces with positive constant Gaussian curvature. The Liebmann theorem is proven showing that the only complete examples must be totally umbilical round spheres. In Sections 4, 5 and 6 the author considers the classification of complete flat surfaces in H3, R3 and S3 respectively. Finally, in Section 7 the author considers the case of complete surfaces with negative constant Gaussian curvature, and he proves the Hilbert theorem. In these notes the author also gives a review on some open questions related to this topic. In particular, there is not much known when we consider the analogous problems for surfaces with constant Gaussian curvature having some certain types of admissible singularities.