A halfspace theorem for proper, negatively curved immersions

Abstract

Generalizing the classical halfspace theorem for minimal surfaces (Hoffman and Meeks in Invent Math 101:373-377, 1990), we prove such a result for two-dimensional surfaces in R 3 of negative Gaussian curvature. Instead of requiring an elliptic differential equation, we merely assume some inequality involving the principal curvatures of the sur- face to be satisfied, see assumption (1). Surfaces of this type arise naturally as critical points of weighted area functionals defined in (2).