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.

中文翻译：

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3 维空间形式中的拟完备 k 曲面

在研究空间形式中恒定正外在曲率的浸没表面时，很自然地用完整性代替较弱的属性，我们在这里称之为准完整性。我们在准完整性的假设下确定了此类表面的全局几何形状。特别是，我们表明，对于$k>\text{最大限度}（0,-C）$，唯一具有恒​​定外在曲率的准完全浸没表面等于$k$在恒定截面曲率的 3 维空间形式中等于$C$是测地线球体。与作者早期的工作一起，这完成了 3 维空间形式中恒定正外曲率的准完全浸入表面的分类。