On constant mean curvature surfaces with periodic metric

TL;DR

This paper studies constant mean curvature surfaces with periodic metrics, showing they are always of finite type and offering a new approach to classifying CMC-tori.

Contribution

It introduces an alternative method for classifying CMC-tori based on periodicity conditions, expanding understanding of CMC surfaces with periodic metrics.

Findings

CMC-surfaces with periodic metric are always of finite type

Develops an alternative classification approach for CMC-tori

Provides insights into the structure of CMC-surfaces in dressing orbits

Abstract

We investigate CMC-surfaces with periodic metric in a dressing orbit of the cylinder. It is shown, that such surfaces are always of finite type. Using the periodicity conditions for the extended frame of a CMC-surface, we develop an alternative approach to the classification of CMC-tori given by Pinkall and Sterling.