The Singly Periodic GenusOne Helicoid
Abstract
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 {\it Bulletin of the AMS}, 29(1):7784, 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 1996
 DOI:
 10.48550/arXiv.math/9605222
 arXiv:
 arXiv:math/9605222
 Bibcode:
 1996math......5222H
 Keywords:

 Mathematics  Differential Geometry