Helicoids with handles and Baker-Akhiezer spinors
type: Item
instance of: scholarly article
title: Helicoids with handles and Baker-Akhiezer spinors
zbMATH Open document ID: 0970.53010
DOI: 10.1007/PL00004652
published in: Mathematische Zeitschrift
publication date: 2000-01-27T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 1229579
zbMATH Keywords: minimal surface
zbMATH Keywords: helicoid
zbMATH Keywords: finite topology
zbMATH Keywords: spinors
zbMATH Keywords: Weierstrass representation
author: Alexander I. Bobenko
reviewed by: Luis J. Alías
MaRDI profile type: Publication
full work available at URL: pl00004652
OpenAlex ID: W2076530442
description: scientific article; zbMATH DE number 1229579