# Commentarii Mathematici Helvetici

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**Volume 74, Issue 2, 1999, pp. 248–279**

**DOI: 10.1007/s000140050088**

The singly periodic genus-one helicoid

D. Hoffman^{[1]}, H. Karcher

^{[2]}and F. Wei

^{[3]}(1) Mathematical Sciences Research Institute, 1000 Centennial Drive, CA 94720-5070, Berkeley, USA

(2) Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

(3) Mathematics Department, Virginia Tech, VA 24061-0123, Blacksburg, USA

We prove the existence of a complete, embedded, singly periodic minimal surface, whose quotient by vertical translations has genus one and two ends. The existence of this surface was announced in our paper in Bulletin of the AMS, 29(1):77-84, 1993. Its ends in the quotient are asymptotic to one full turn of the helicoid, and, like the helicoid, it contains a vertical line. Modulo vertical translations, it has two parallel horizontal lines crossing the vertical axis. The nontrivial symmetries of the surface, modulo vertical translations, consist of: $ 180^\circ $-rotation about the vertical line; $ 180^\circ $ rotation about the horizontal lines (the same symmetry); and their composition.

*Keywords: *Minimal surface, embedded, elliptic functions, Riemann surfaces

Hoffman D., Karcher H., Wei F.: The singly periodic genus-one helicoid. *Comment. Math. Helv.* 74 (1999), 248-279. doi: 10.1007/s000140050088