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

TL;DR

This paper classifies quasicomplete surfaces with constant positive extrinsic curvature in 3D space-forms, showing they are geodesic spheres under certain conditions, thus completing the classification of such surfaces.

Contribution

It introduces the concept of quasicompleteness and proves that quasicomplete surfaces with certain curvature conditions are geodesic spheres, extending previous classifications.

Findings

Quasicomplete surfaces with constant extrinsic curvature are geodesic spheres.

The classification is complete for surfaces with curvature greater than Max(0, -c).

This work extends the understanding of immersed surface geometries in space-forms.

Abstract

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.