On quasicomplete k$k$‐surfaces in 3‐dimensional space‐forms

Abstract

AbstractIn the study of immersed surfaces of constant positive extrinsic curvature in space‐forms, it is natural to substitute completeness for a weaker property, which we here call quasicompleteness. We determine the global geometry of such surfaces under the hypotheses of quasicompleteness. In particular, we show that, for , the only quasicomplete immersed surfaces of constant extrinsic curvature equal to in the 3‐dimensional space‐form of constant sectional curvature equal to are the geodesic spheres. Together with earlier work of the author, this completes the classification of quasicomplete immersed surfaces of constant positive extrinsic curvature in 3‐dimensional space‐forms.