Complete surfaces with ends of non positive curvature. (English) Zbl 1319.53006
Summary: The classical Efimov theorem states that there is no \(\mathcal{C}^2\)-smoothly immersed complete surface in \(\mathbb{R}^3\) with negative Gauss curvature uniformly separated from zero. Here we analyze the case when the curvature of the complete surface is less that \(- c^2\) in a neighborhood of infinity, and prove the surface is topologically a finitely punctured compact surface, the area is finite, and each puncture looks like cusps extending to infinity, asymptotic to rays.
MSC:
| 53A07 | Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces |
| 49Q10 | Optimization of shapes other than minimal surfaces |
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