Soliton spheres

Bohle, Christoph; Peters, G. Paul

doi:10.1090/S0002-9947-2011-05323-7

Soliton spheres
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by Christoph Bohle and G. Paul Peters PDF

Trans. Amer. Math. Soc. 363 (2011), 5419-5463 Request permission

Abstract:

Soliton spheres are immersed 2–spheres in the conformal 4–sphere $S^4=\mathbb {HP}^1$ that allow rational, conformal parametrizations $f\colon \mathbb {CP}^1\rightarrow \mathbb {HP}^1$ obtained via twistor projection and dualization from rational curves in $\mathbb {CP}^{2n+1}$. Soliton spheres can be characterized as the case of equality in the quaternionic Plücker estimate. A special class of soliton spheres introduced by Taimanov are immersions into $\mathbb {R}^3$ 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 $W=4\pi d$ with $d\in \mathbb {N}\backslash \{0,2,3,5,7\}$. The same quantization was previously known individually for each of the three special classes of soliton spheres mentioned above.

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