Helicoids with handles and Baker-Akhiezer spinors

Abstract

For more then 200 years the helicoid was the only known infinite total curvature embedded minimal surface of finite topology. The situation changed in 1993, when Hoffman, Karcher and Wei [9] discovered the genus one helicoid a minimal torus with one end, which has a form of the helicoid at infinity. The genus one helicoid was constructed using the Weierstrass representation. Karcher, Wei and Hoffman have solved the corresponding period problem [11] and were able to produce detailed plots of the surface, strongly suggesting that it is embedded. The Gauss map of this surface has an essential singularity at the puncture. This makes the problem familiar to specialists in the theory of integrable systems, where the Baker-Akhiezer functions – functions with essential singularities on compact Riemann surfaces – have become a basic tool of the finite-gap integration theory (see for example [13], [1]). In the present paper we describe all immersed minimal surfaces of finite topology with just one helicoidal end. Using the spinor Weierstrass representation [19,3,15] these immersions are described in terms of holomorphic spinors with essential singularities at the puncture, which we call the BakerAkhiezer spinors. Those are described explicitly, as well as the Gauss map, which is a meromorphic function with an essential singularity at the puncture. Further we discuss the periodicity conditions for the immersion and show how they yield the Riemann surfaces, which are two-sheeted ramified coverings. This motivates the following