1998/01/08 by Iskander A. Taimanov
Mathematics · #math.DG
published as Proceedings of the Steklov Institute of Math. 225 (1999), 222-243. · 30 pages, LaTeX; Preprint no. 302, SFB 288, TU-Berlin
arxiv created 1998/01/08 · arxiv updated 2009/11/30
The Weierstrass representation for spheres in \R3 and, in particular, effective construction of immersions from data of spectral theory origin is discussed. These data are related to Dirac operators on a plane and on an infinite cylinder and these operators are just representations of Dirac operators acting in spinor bundles over the two-sphere which is naturally obtained as a completion of a plane or of a cylinder. Spheres described in terms of Dirac operators with one-dimensional potentials on a cylinder are completely studied and, in particular, for them a lower estimate of the Willmore functional in terms of the dimension of the kernel of the corresponding Dirac operator on a two-sphere is obtained. It is conjectured that this estimate is valid for all Dirac operators on spheres and some reasonings for this conjecture are discussed. In Appendix a criterion distinguishing Weierstrass representations, of universal coverings of compact surfaces of higher genera, converted into immersions of compact surfaces is given.