Some Restrictions on the Smooth Immersion of Complete Surfaces in E3

Abstract

This chapter presents an the idea of Hawley in direction of conjecture that says if S be a complete, umbilic-free surface, C2 immersed in E3, so that the sum of the absolute values of the principal curvatures is bounded away from zero. Then Gauss curvature K on S either changes sign or else vanishes identically. A theorem due to Hawley that says if S be a complete, antiminimal, 2-bounded surface, C3 imbedded in E3 with K ≤ 0, then S is a cylinder over an oval. The chapter presents the generalization of Hawley's theorem.