The dynamics theorem for CMC surfaces in R^3

William H. Meeks, Giuseppe Tinaglia

Research output: Contribution to journalArticlepeer-review


In this paper, we study the space of translational limits T(M) of a surface M properly embedded in R-3 with nonzero constant mean curvature and bounded second fundamental form. There is a natural map T which assigns to any surface Sigma is an element of T(M) the set T(Sigma) subset of 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.
Original languageEnglish
Pages (from-to)141 - 173
Number of pages33
JournalJournal of Differential Geometry
Issue number1
Publication statusPublished - 2010


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