The embedded Calabi-Yau conjecture for finite genus. (English) Zbl 1490.53014
The paper is an important step in the study of the embedded Calabi-Yau conjecture for finite genus which states: every connected, complete embedded minimal surface \(M\subset\mathbb{R}^3\) of finite genus and compact (possibly empty) boundary is properly embedded in \(\mathbb{R}^3\). Among special cases of the conjecture, if \(M\) has finite topology, see for exemple [T. H. Colding and W. P. Minicozzi, Ann. Math. (2) 167, No. 1, 211–243 (2008; Zbl 1142.53012)] or if \(M\) has positive injectivity radius [W. H. Meeks III and H. Rosenberg, Duke Math. J. 133, No. 3, 467–497 (2006; Zbl 1098.53007)].
The authors consider an \(M\) as in the conjecture above with an infinite number of ends. They show that a simple limit end of \(M\) has a properly embedded representative with compact boundary and genus zero and they give detailed information on its geometry. They derive that if \(M\) has at least 2 simple limit ends, it has exactly 2 simple limit ends.
They prove that \(M\) is properly embedded in \(\mathbb{R}^3\) if and only if it has a countable number of limit ends; and they show that in that case, \(M\) has actually at most 2 limit ends.
The authors consider an \(M\) as in the conjecture above with an infinite number of ends. They show that a simple limit end of \(M\) has a properly embedded representative with compact boundary and genus zero and they give detailed information on its geometry. They derive that if \(M\) has at least 2 simple limit ends, it has exactly 2 simple limit ends.
They prove that \(M\) is properly embedded in \(\mathbb{R}^3\) if and only if it has a countable number of limit ends; and they show that in that case, \(M\) has actually at most 2 limit ends.
Reviewer: Marina Ville (Paris)
MSC:
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
| 49Q05 | Minimal surfaces and optimization |
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
Keywords:
embedded Calabi-Yau problem; injectivity radius function; limit end; locally simply connected; minimal lamination; proper minimal surfaceReferences:
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