Triply periodic constant mean curvature surfaces

Triply periodic minimal and constant mean curvature surfaces.

We want to summarize some established results on periodic surfaces which are minimal or have constant mean curvature, along with some recent results. We will do this from a mathematical point of view with a general readership in mind.

Stable periodic constant mean curvature surfaces and mesoscopic phase separation

We give a first comprehensive description of the stable solutions of the periodic isoperimetric problem in the case of lattice symmetry. This result is intended to elucidate the geometry of certain sophisticated interfaces appearing in mesoscale phase separation phenomena. We prove that closed, stable, constant mean curvature surfaces in R3/Γ , Γ ⊂ R3 being a discrete subgroup of translations w...

New Constant Mean Curvature Surfaces

We use the DPW construction [5] to present three new classes of immersed CMC cylinders, each of which includes surfaces with umbilics. The first class consists of cylinders with one end asymptotic to a Delaunay surface. The second class presents surfaces with a closed planar geodesic. In the third class each surface has a closed curve of points with a common tangent plane. An appendix, by the t...

Coplanar Constant Mean Curvature Surfaces

We consider constant mean curvature surfaces with finite topology, properly embedded in three-space in the sense of Alexandrov. Such surfaces with three ends and genus zero were constructed and completely classified by the authors [GKS2, GKS1]. Here we extend the arguments to the case of an arbitrary number of ends, under the assumption that the asymptotic axes of the ends lie in a common plane...

Surfaces of Constant Mean Curvature