The Singly Periodic Genus-One 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):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.
- Publication:
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arXiv Mathematics e-prints
- Pub Date:
- May 1996
- DOI:
- 10.48550/arXiv.math/9605222
- arXiv:
- arXiv:math/9605222
- Bibcode:
- 1996math......5222H
- Keywords:
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- Mathematics - Differential Geometry