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The Weierstrass representation of spheres in $R^3$, the Willmore numbers, and soliton spheres

Iskander A. Taimanov
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Abstract

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 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.

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Iskander A. Taimanov. 1998-01-08. The Weierstrass representation of spheres in $R^3$, the Willmore numbers, and soliton spheres. https://arxiv.org/abs/math/9801022

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