Soliton Spheres

TL;DR

Soliton spheres are special conformal 2-spheres in S^4 characterized by rational parametrizations, related to integrable systems, and include notable examples like Willmore and Bryant spheres, with quantized Willmore energy values.

Contribution

This paper characterizes soliton spheres via the quaternionic Pluecker estimate, relates them to integrable systems, and classifies their Willmore energy quantization.

Findings

Soliton spheres can be described through rational conformal parametrizations from twistor projections.

Willmore and Bryant spheres with smooth ends are examples of soliton spheres.

The Willmore energy of soliton spheres in S^3 is quantized as 4pi times a positive integer, excluding 2, 3, 5, and 7.

Abstract

Soliton spheres are immersed 2-spheres in the conformal 4-sphere S^4=HP^1 that allow rational, conformal parametrizations f:CP^1->HP^1 obtained via twistor projection and dualization from rational curves in CP^{2n+1}. Soliton spheres can be characterized as the case of equality in the quaternionic Pluecker estimate. A special class of soliton spheres introduced by Taimanov are immersions into R^3 with rotationally symmetric Weierstrass potentials that are related to solitons of the mKdV-equation via the ZS-AKNS linear problem. We show that Willmore spheres and Bryant spheres with smooth ends are further examples of soliton spheres. The possible values of the Willmore energy for soliton spheres in the 3-sphere are proven to be W=4pi*d with d a positive integer but not 2,3,5, or 7. The same quantization was previously known individually for each of the three special classes of soliton spheres mentioned above.