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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Soliton spheres


Authors: Christoph Bohle and G. Paul Peters
Journal: Trans. Amer. Math. Soc. 363 (2011), 5419-5463
MSC (2000): Primary 53C42; Secondary 53A30, 37K25
Published electronically: May 24, 2011
MathSciNet review: 2813421
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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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Additional Information

Christoph Bohle
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
Email: bohle@mathematik.uni-tuebingen.de

G. Paul Peters
Affiliation: Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany
Email: peters@math.tu-berlin.de

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05323-7
PII: S 0002-9947(2011)05323-7
Received by editor(s): May 20, 2009
Received by editor(s) in revised form: January 8, 2010
Published electronically: May 24, 2011
Additional Notes: Both authors were supported by DFG SPP 1154 “Global Differential Geometry”.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.