The dynamics theorem for CMC surfaces in $R^3$

Abstract

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 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. Mathematics Subject Classification: Primary 53A10, Secondary 49Q05, 53C42

Limit lamination theorems for H-surfaces

“…and the sequence has locally bounded norm of the second fundamental form in R 3 . By a standard argument, a subsequence of the surfaces converges to a weak H-lamination L of R 3 ; see the references [20,25,32] for this argument and the Appendix for the definition and some key properties of a weak H-lamination that we will apply below.…”

“…For further background material on these notions see Section 3 of [18], [25] or our previous papers [32,33]. Definition 6.1.…”

Constant mean curvature surfaces

“…The material covered here is based on [149] by Meeks and Tinaglia, which was motivated by the earlier work in [122] and we refer the reader to [149] for further details. We will focus our attention here on some of the less technical results in [149] and the basic techniques developed there.…”

“…In Sections 10 and 11 we cover some results of Meeks and Tinaglia mentioned previously, as well as their Dynamics and Minimal Elements Theorems for complete strongly Alexandrov embedded 1-surfaces in R 3 from [149]. This Minimal Elements Theorem is needed in the proofs of the curvature and radius estimates stated previously in Theorems 1.6 and 1.7.…”

Curvature estimates for constant mean curvature surfaces

“…A standard compactness argument, see for instance Section 2.1 in this manuscript or the paper [24], gives that a subsequence of ∆ n converges with multiplicity one to a genus zero, strongly Alexandrov embedded 1 1-surface ∆ ∞ with bounded norm of the second fundamental form. By the Minimal Element Theorem in [24], for some divergent sequence of points q n ∈ ∆ ∞ , the translated surfaces ∆ ∞ − q n converge with multiplicity one to a strongly Alexandrov embedded surface ∆ ∞ in R 3 such that the component passing through 0 is an embedded Delaunay surface. Since a Delaunay surface has nonzero flux, we conclude that the original disks D n also have nonzero flux for n large, which is a contradiction.…”