The Dynamics Theorem for CMC surfaces in R^3

William H. Meeks, III University of Massachusetts Giuseppe Tinaglia King’s College London

Differential Geometry mathscidoc:1609.10180

Journal of Differential Geometry, 85, (1), 141-173, 2010
In this paper, we study the space of translational limits T (M) of a surface M properly embedded in R3 with nonzero constant mean curvature and bounded second fundamental form. There is a natural map T which assigns to any surface Σ ∈ T (M) the set T (Σ) ⊂ T(M). Among various dynamics type results we prove that surfaces in minimal T -invariant sets of T (M) are chord-arc. We also show that if M has an infinite number of ends, then there exists a nonempty minimal T -invariant set in T (M) consisting entirely of surfaces with planes of Alexandrov symmetry. Finally, when M has a plane of Alexandrov symmetry, we prove the following characterization theorem: M has finite topology if and only if M has a finite number of ends greater than one.
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  title={The Dynamics Theorem for CMC surfaces in R^3},
  author={William H. Meeks, III, and Giuseppe Tinaglia},
  booktitle={Journal of Differential Geometry},
William H. Meeks, III, and Giuseppe Tinaglia. The Dynamics Theorem for CMC surfaces in R^3. 2010. Vol. 85. In Journal of Differential Geometry. pp.141-173.
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