The Dynamics Theorem for CMC surfaces in R^3

TL;DR

This paper investigates the dynamics of constant mean curvature surfaces in R^3, revealing conditions for finite topology, symmetry, and the structure of their translational limits.

Contribution

It introduces new results on the structure of T-invariant sets, symmetry properties, and topological characterizations of CMC surfaces in Euclidean space.

Findings

Surfaces in minimal T-invariant sets are chord-arc.

Existence of T-invariant sets with surfaces having planes of Alexandrov symmetry.

Characterization of finite topology based on the number of ends when a plane of symmetry exists.

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 M' in T(M), the set T(M') in 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.