On constant mean curvature surfaces with periodic metric (Q1392490)

The authors reproduce the classification of constant mean curvature (CMC) tori using extended frames without referring to the metric (i.e., the sinh-Gordon equation) explicitly. They derive a classification of all finite type surfaces with periodic metric, i.e., surfaces whose metric is a finite type solution of the sinh-Gordon equation which is invariant under a group of translations in \(C\). The authors use the DPW method rather than the approach of \textit{U. Pinkall} and \textit{I. Sterling} [Ann. Math., II. Ser. 130, 407-451 (1989; Zbl 0683.53053)] which used the sinh-Gordon equation. The authors provide a short description of the DPW method, discuss translational symmetries as well as hyperelliptic curves, and conclude by giving an algebro-geometric description of surfaces with periodic metrics.