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Abstract: In this paper, we study the space of translational limits TM of a surface Mproperly embedded in R^3 with nonzero constant mean curvature and boundedsecond fundamental form. There is a natural map T which assigns to any surfaceM- in TM, the set TM- in TM. Among various dynamics type results we provethat surfaces in minimal T-invariant sets of TM are chord-arc. We also showthat if M has an infinite number of ends, then there exists a nonempty minimalT-invariant set in TM consisting entirely of surfaces with planes ofAlexandrov symmetry. Finally, when M has a plane of Alexandrov symmetry, weprove the following characterization theorem: M has finite topology if and onlyif M has a finite number of ends greater than one.



Author: William H. Meeks III, Giuseppe Tinaglia

Source: https://arxiv.org/







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