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Sfb 288 Differential Geometry and Quantum Physics |
Abstract for Sfb Preprint No. 302
The Weierstrass representation of spheres in <B>R</B>^3, the Willmore numbers, and soliton spheres I. A. Taimanov
The Weierstrass representation for spheres in $R^3$ and, in particular,effective construction of immersions from data of spectral theory origin is discussed. These data are related to Dirac operatorson a plane and on an infinite cylinder and these operators are justrepresentations of Dirac operators acting in spinor bundles over the two-sphere which is naturally obtained as a completion of a plane or ofa cylinder. Spheres described in terms of Dirac operators with one-dimensional potentials on a cylinder are completely studied and, in particular, for them alower estimate of the Willmore functional in terms of the dimensionof 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.
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