Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 264

Helicoids with handles and Baker-Akhiezer spinors

A. I. Bobenko

Math. Z. 229 (1998) 9-29

All immersed minimal surfaces of finite topology with one helicoidal end are described. Using the spinor Weierstrass representation these immersions are described in terms of holomorphic spinors with essential singularities at the puncture, which we call the Baker-Akhiezer spinors. Those are described explicitly in terms of the Riemann theta functions. The analysis of the periodicity conditions strongly suggests that any immersed minimal surface of finite topology with one helicoidal end has normal symmetry: it is invariant with respect to a 180^0-rotation about a line orthogonal to the surface. In particular a new description of the Karcher-Hoffman-Wei genus one helicoid is presented.

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