Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-2011-05323-7
Published electronically: May 24, 2011
MathSciNet review: 2813421
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. Uwe Abresch. Constant mean curvature tori in terms of elliptic functions. J. Reine Angew. Math. 374 (1987), 169-192. MR 876223 (88e:53006)
  • 2. Alexander I. Bobenko. Surfaces of constant mean curvature and integrable equations. (Russian) Uspekhi Mat. Nauk 46 (1991), no. 4 (280), 3-42, 192; translation in Russian Math. Surveys 46 (1991), no. 4, 1-45. MR 1138951 (93b:53009)
  • 3. Christoph Bohle, Constrained Willmore tori in the $ 4$-sphere. J. Diff. Geom. 86 (2010), 71-131.
  • 4. Christoph Bohle, Katrin Leschke, Franz Pedit, and Ulrich Pinkall, Conformal maps from a 2-torus to the 4-sphere. To appear in J. Reine Angew. Math.
  • 5. Christoph Bohle and G. Paul Peters. Bryant surfaces with smooth ends. Comm. Anal. Geom. 17 (2009), 587-619. MR 2601345
  • 6. Robert L. Bryant. Conformal and minimal immersions of compact surfaces into the $ 4$-sphere. J. Differential Geom. 17 (1982), no. 3, 455-473. MR 679067 (84a:53062)
  • 7. Robert L. Bryant. A duality theorem for Willmore surfaces. J. Differential Geom. 20 (1984), no. 1, 23-53. MR 772125 (86j:58029)
  • 8. Robert L. Bryant. Surfaces of mean curvature one in hyperbolic space. Théorie des variétés minimales et applications (Palaiseau, 1983-1984). Astérisque No. 154-155 (1987), 12, 321-347, 353 (1988). MR 955072
  • 9. Robert L. Bryant. Surfaces in conformal geometry. The mathematical heritage of Hermann Weyl (Durham, NC, 1987), 227-240, Proc. Sympos. Pure Math., 48, Amer. Math. Soc., Providence, RI, 1988. MR 974338 (89m:53102)
  • 10. Francis E. Burstall, Dirk Ferus, Katrin Leschke, Franz Pedit, and Ulrich Pinkall. Conformal Geometry of Surfaces in $ S^4$ and Quaternions, Lecture Notes in Mathematics 1772, Springer, Berlin, 2002. MR 1887131 (2004a:53058)
  • 11. Celso J. Costa. Example of a complete minimal immersion in $ \mathbb{R}\sp 3$ of genus one and three embedded ends. Bol. Soc. Brasil. Mat. 15 (1984), no. 1-2, 47-54. MR 794728 (87c:53111)
  • 12. Norio Ejiri. Willmore surfaces with a duality in $ S\sp N(1)$. Proc. London Math. Soc. (3) 57 (1988), no. 2, 383-416. MR 950596 (89h:53117)
  • 13. Dirk Ferus, Katrin Leschke, Franz Pedit, and Ulrich Pinkall. Quaternionic holomorphic geometry: Plücker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori. Invent. Math. 146 (2001), 507-593. MR 1869849 (2003a:53057)
  • 14. Thomas Friedrich. On surfaces in four-space. Ann. Global Anal. Geom. 2 (1984), no. 3, 257-287. MR 777909 (86h:53061)
  • 15. Thomas Friedrich. The geometry of $ t$-holomorphic surfaces in $ S^4$. Math. Nachr. 137 (1988),49-62. MR 968986 (90a:53075)
  • 16. Thomas Friedrich. On superminimal surfaces. Arch. Math. (Brno) 33 (1997), no. 1-2, 41-56. MR 1464300 (98h:53092)
  • 17. Philipp Griffith and Joseph Harris. Principles of algebraic geometry. Pure and applied mathematics, John Wiley & Sons, New York, 1978, Wiley Classics Library Edition, 1994. MR 507725 (80b:14001)
  • 18. Robert D. Gulliver II, Robert Osserman, and Halsey L. Royden. A theory of branched immersions of surfaces. Amer. J. Math. 95 (1973), 750-812. MR 0362153 (50:14595)
  • 19. Udo Hertrich-Jeromin, Emilio Musso, and Lorenzo Nicolodi, Möbius geometry of surfaces of constant mean curvature 1 in hyperbolic space. Ann. Global Anal. Geom. 19 (2001), no. 2, 185-205. MR 1826401 (2002a:53079)
  • 20. Udo Hertrich-Jeromin. Introduction to Möbius Differential Geometry. Lecture Notes Series 300, Cambridge University Press, 2003. MR 2004958 (2004g:53001)
  • 21. Nigel J. Hitchin. Harmonic maps from a $ 2$-torus to the $ 3$-sphere. J. Differential Geom. 31 (1990), no. 3, 627-710. MR 1053342 (91d:58050)
  • 22. David A. Hoffman and William H. Meeks III. A complete embedded minimal surface in $ \mathbb{R}\sp 3$ with genus one and three ends. J. Differential Geom. 21 (1985), no. 1, 109-127. MR 806705 (87d:53008)
  • 23. Boris Konopelchenko. Induced surfaces and their integrable dynamics. Stud. Appl. Math. 96 (1996), no. 1, 9-51. MR 1365273 (96i:53011)
  • 24. William H. Meeks, Antonio Ros, and Herold Rosenberg. The global theory of minimal surfaces in flat spaces. Lectures given at the 2nd C.I.M.E. Session held in Martina Franca, July 7-14, 1999. Edited by Gian Pietro Pirola. Lecture Notes in Mathematics, 1775. Springer-Verlag, Berlin, 2002. MR 1901611 (2003i:53012)
  • 25. Sebastián Montiel. Willmore two-spheres in the four-sphere. Trans. Amer. Math. Soc. 352 (2000), no. 10, 4469-4486. MR 1695032 (2001b:53080)
  • 26. Emilio Musso. Willmore surfaces in the four-sphere. Ann. Global Anal. Geom. 8 (1990), no. 1, 21-41. MR 1075237 (92g:53059)
  • 27. Robert A. Osserman. A survey of minimal surfaces. Second edition. Dover Publications, Inc., New York, 1986. MR 852409 (87j:53012)
  • 28. Franz Pedit and Ulrich Pinkall. Quaternionic analysis on Riemann surfaces and differential geometry. Doc. Math. J. DMV, Extra Volume ICM Berlin 1998, Vol. II, 389-400. MR 1648089 (2000c:53053)
  • 29. G. Paul Peters. Soliton Spheres. Thesis, TU-Berlin, http://nbn-resolving.de/urn:nbn:de:kobv:83-opus-8422urn:nbn:de:kobv:83-opus-8422, 2004.
  • 30. Ulrich Pinkall and Ivan Sterling. On the classification of constant mean curvature tori. Ann. of Math. (2) 130 (1989), no. 2, 407-451. MR 1014929 (91b:53009)
  • 31. Jörg Richter. Conformal maps of a Riemann surface into the space of quaternions. Thesis, TU-Berlin, 1997.
  • 32. Iskander A. Taimanov. Modified Novikov-Veselov equation and differential geometry of surfaces. Solitons, geometry, and topology: on the crossroad, 133-151, Amer. Math. Soc. Transl. Ser. 2, 179, Amer. Math. Soc., Providence, RI, 1997. MR 1437161 (98c:53071)
  • 33. Iskander A. Taimanov. The Weierstrass representation of closed surfaces in $ R\sp 3$. (Russian) Funktsional. Anal. i Prilozhen. 32 (1998), no. 4, 49-62, 96; translation in Funct. Anal. Appl. 32 (1998), no. 4, 258-267. MR 1678856 (2000d:53006)
  • 34. Iskander A. Taimanov. The Weierstrass representation of spheres in $ \mathbb{R}^3$, the Willmore numbers, and soliton spheres. Tr. Mat. Inst. Steklova, 225 (1999), Solitony Geom. Topol. na Perekrest., 339-361; translation in Proc. Steklov Inst. Math. (1999), no. 2 225, 322-343. MR 1725951 (2001g:37135)
  • 35. Iskander A. Taimanov. The two-dimensional Dirac operator and the theory of surfaces. (Russian) Uspekhi Mat. Nauk 61 (2006), no. 1(367), 85-164; translation in Russian Math. Surveys 61 (2006), no. 1, 79-159. MR 2239773 (2007k:37098)
  • 36. Henry C. Wente. Counterexample to a conjecture of H. Hopf. Pacific J. Math. 121 (1986), no. 1, 193-243. MR 815044 (87d:53013)
  • 37. Thomas J. Willmore, Riemannian geometry. Oxford University Press, Oxford, New York, 1993. MR 1261641 (95e:53002)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53C42, 53A30, 37K25

Retrieve articles in all journals with MSC (2000): 53C42, 53A30, 37K25


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: https://doi.org/10.1090/S0002-9947-2011-05323-7
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.

American Mathematical Society