The Chen-Gackstatter surface, discovered 1982 by Chi Chen Cheng and Fritz Gackstatter, was the first interesting complete minimal torus of finite total curvature (it was known before that there were such surfaces with extremely complicated ends).

The surface has, like the Enneper surface, two horizontal straight lines and two vertical planes of symmetry. The additional handle causes what is called the period problem. If you don’t take care of it, the surface will have gaps.

The symmetries imply that the underlying torus is a square torus. The points with vertical normals occur at the 2-division points of this torus. There, the Gauss map G will have simple poles/zeroes, while the height differential dh will have three simple zeroes at the finite points and a triple order pole at the end. This essentially determines G and dh up to a scale factor, which is adjusted so that the period problem is solved. This can be visualized by looking at one quarter of the flat structures of G dh and 1/G dh. They have to be scaled so that they fit together:



Reference: C.C. Chen and F. Gackstatter: Elliptische und hyperelliptische Funktionen und vollständige Minimalflächen vom Enneperschen Typ, Math. Ann. 259, 359-369, 1982.

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