The dynamics theorem for CMC surfaces in R^3

TY - JOUR

T1 - The dynamics theorem for CMC surfaces in R^3

AU - Meeks, William H.

AU - Tinaglia, Giuseppe

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

M3 - Article

VL - 85

SP - 141

EP - 173

JO - Journal of Differential Geometry

JF - Journal of Differential Geometry

IS - 1

ER -