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

In 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 $k>\text{Max}(0,-c)$, the only quasicomplete immersed surfaces of constant extrinsic curvature equal to $k$ in the $3$-dimensional space-form of constant sectional curvature equal to $c$ 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.