Helicoids with handles and Baker-Akhiezer spinors
For more than 200 years the helicoid was the only known example of an embedded minimal surface of finite topology and infinite total curvature. The situation changed in 1993, when \textit{D. Hoffman, H. Karcher} and \textit{F. Wei} [Comment. Math. Helv. 74, 248-279 (1999; Zbl 0958.53006)] discovered the genus one helicoid, a minimal torus with one end, which has the form of the helicoid at infinity. The genus one helicoid was constructed using the Weierstrass representation, and its plots strongly suggested that it was embedded. In the paper under review the author describes all immersed minimal surfaces of finite topology with just one helicoidal end. Using the spinor Weierstrass representation, these immersions are described in terms of Baker-Akhiezer spinors, which are holomorphic spinors with essential singularities at the puncture. Further the author discusses the periodicity conditions for the immersion and shows how they yield the Riemann surfaces, which are two-sheeted ramified coverings. Motivated by this, the author conjectures that any immersed minimal surface of finite topology with one helicoidal end has normal symmetry, that is, it is invariant with respect to a 180 degree rotation about a line orthogonal to the surface. In the last section of the paper, the author presents a new description of the Karcher-Hoffman-Wei genus one helicoid, which contains only one integration. Choosing a different spinor structure, the author constructs a twisted genus one helicoid. The paper also contains plots of the genus one helicoids produced by the author's student W. Hornauer.
scientific article; zbMATH DE number 1229579