Soliton spheres
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- by Christoph Bohle and G. Paul Peters PDF
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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.References
- U. Abresch, Constant mean curvature tori in terms of elliptic functions, J. Reine Angew. Math. 374 (1987), 169–192. MR 876223, DOI 10.1515/crll.1987.374.169
- A. I. Bobenko, Surfaces of constant mean curvature and integrable equations, Uspekhi Mat. Nauk 46 (1991), no. 4(280), 3–42, 192 (Russian); English transl., Russian Math. Surveys 46 (1991), no. 4, 1–45. MR 1138951, DOI 10.1070/RM1991v046n04ABEH002826
- Christoph Bohle, Constrained Willmore tori in the $4$–sphere. J. Diff. Geom. 86 (2010), 71–131.
- 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.
- Christoph Bohle and G. Paul Peters, Bryant surfaces with smooth ends, Comm. Anal. Geom. 17 (2009), no. 4, 587–619. MR 2601345, DOI 10.4310/CAG.2009.v17.n4.a1
- Robert L. Bryant, Conformal and minimal immersions of compact surfaces into the $4$-sphere, J. Differential Geometry 17 (1982), no. 3, 455–473. MR 679067
- Robert L. Bryant, A duality theorem for Willmore surfaces, J. Differential Geom. 20 (1984), no. 1, 23–53. MR 772125
- Robert L. Bryant, Surfaces of mean curvature one in hyperbolic space, Astérisque 154-155 (1987), 12, 321–347, 353 (1988) (English, with French summary). Théorie des variétés minimales et applications (Palaiseau, 1983–1984). MR 955072
- Robert L. Bryant, Surfaces in conformal geometry, The mathematical heritage of Hermann Weyl (Durham, NC, 1987) Proc. Sympos. Pure Math., vol. 48, Amer. Math. Soc., Providence, RI, 1988, pp. 227–240. MR 974338, DOI 10.1090/pspum/048/974338
- F. E. Burstall, D. Ferus, K. Leschke, F. Pedit, and U. Pinkall, Conformal geometry of surfaces in $\textit {S}^4$ and quaternions, Lecture Notes in Mathematics, vol. 1772, Springer-Verlag, Berlin, 2002. MR 1887131, DOI 10.1007/b82935
- Celso J. Costa, Example of a complete minimal immersion in $\textbf {R}^3$ of genus one and three embedded ends, Bol. Soc. Brasil. Mat. 15 (1984), no. 1-2, 47–54. MR 794728, DOI 10.1007/BF02584707
- Norio Ejiri, Willmore surfaces with a duality in $S^N(1)$, Proc. London Math. Soc. (3) 57 (1988), no. 2, 383–416. MR 950596, DOI 10.1112/plms/s3-57.2.383
- D. Ferus, K. Leschke, F. Pedit, and U. Pinkall, Quaternionic holomorphic geometry: Plücker formula, Dirac eigenvalue estimates and energy estimates of harmonic $2$-tori, Invent. Math. 146 (2001), no. 3, 507–593. MR 1869849, DOI 10.1007/s002220100173
- Thomas Friedrich, On surfaces in four-spaces, Ann. Global Anal. Geom. 2 (1984), no. 3, 257–287. MR 777909, DOI 10.1007/BF01876417
- Thomas Friedrich, The geometry of $t$-holomorphic surfaces in $S^4$, Math. Nachr. 137 (1988), 49–62. MR 968986, DOI 10.1002/mana.19881370106
- Thomas Friedrich, On superminimal surfaces, Arch. Math. (Brno) 33 (1997), no. 1-2, 41–56. MR 1464300
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
- R. D. Gulliver II, R. Osserman, and H. L. Royden, A theory of branched immersions of surfaces, Amer. J. Math. 95 (1973), 750–812. MR 362153, DOI 10.2307/2373697
- U. Hertrich-Jeromin, E. Musso, and L. 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, DOI 10.1023/A:1010738712475
- Udo Hertrich-Jeromin, Introduction to Möbius differential geometry, London Mathematical Society Lecture Note Series, vol. 300, Cambridge University Press, Cambridge, 2003. MR 2004958, DOI 10.1017/CBO9780511546693
- N. J. Hitchin, Harmonic maps from a $2$-torus to the $3$-sphere, J. Differential Geom. 31 (1990), no. 3, 627–710. MR 1053342, DOI 10.4310/jdg/1214444631
- David A. Hoffman and William Meeks III, A complete embedded minimal surface in $\textbf {R}^3$ with genus one and three ends, J. Differential Geom. 21 (1985), no. 1, 109–127. MR 806705
- B. G. Konopelchenko, Induced surfaces and their integrable dynamics, Stud. Appl. Math. 96 (1996), no. 1, 9–51. MR 1365273, DOI 10.1002/sapm19969619
- W. H. Meeks III, A. Ros, and H. Rosenberg, The global theory of minimal surfaces in flat spaces, Lecture Notes in Mathematics, vol. 1775, Springer-Verlag, Berlin; Centro Internazionale Matematico Estivo (C.I.M.E.), Florence, 2002. Lectures given at the 2nd C.I.M.E. Session held in Martina Franca, July 7–14, 1999; Edited by Gian Pietro Pirola; Fondazione CIME/CIME Foundation Subseries. MR 1901611
- Sebastián Montiel, Willmore two-spheres in the four-sphere, Trans. Amer. Math. Soc. 352 (2000), no. 10, 4469–4486. MR 1695032, DOI 10.1090/S0002-9947-00-02571-X
- Emilio Musso, Willmore surfaces in the four-sphere, Ann. Global Anal. Geom. 8 (1990), no. 1, 21–41. MR 1075237, DOI 10.1007/BF00055016
- Robert Osserman, A survey of minimal surfaces, 2nd ed., Dover Publications, Inc., New York, 1986. MR 852409
- Franz Pedit and Ulrich Pinkall, Quaternionic analysis on Riemann surfaces and differential geometry, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 389–400. MR 1648089
- G. Paul Peters. Soliton Spheres. Thesis, TU-Berlin, urn:nbn:de:kobv:83-opus-8422, 2004.
- U. Pinkall and I. Sterling, On the classification of constant mean curvature tori, Ann. of Math. (2) 130 (1989), no. 2, 407–451. MR 1014929, DOI 10.2307/1971425
- Jörg Richter. Conformal maps of a Riemann surface into the space of quaternions. Thesis, TU–Berlin, 1997.
- Iskander A. Taimanov, Modified Novikov-Veselov equation and differential geometry of surfaces, Solitons, geometry, and topology: on the crossroad, Amer. Math. Soc. Transl. Ser. 2, vol. 179, Amer. Math. Soc., Providence, RI, 1997, pp. 133–151. MR 1437161, DOI 10.1090/trans2/179/07
- I. A. Taĭmanov, The Weierstrass representation of closed surfaces in $\textbf {R}^3$, Funktsional. Anal. i Prilozhen. 32 (1998), no. 4, 49–62, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 32 (1998), no. 4, 258–267 (1999). MR 1678856, DOI 10.1007/BF02463208
- I. A. Taĭmanov, The Weierstrass representation of spheres in $\mathbf R^3$, Willmore numbers, and soliton spheres, Tr. Mat. Inst. Steklova 225 (1999), no. Solitony Geom. Topol. na Perekrest., 339–361 (Russian); English transl., Proc. Steklov Inst. Math. 2(225) (1999), 322–343. MR 1725951
- I. A. Taĭmanov, The two-dimensional Dirac operator and the theory of surfaces, Uspekhi Mat. Nauk 61 (2006), no. 1(367), 85–164 (Russian, with Russian summary); English transl., Russian Math. Surveys 61 (2006), no. 1, 79–159. MR 2239773, DOI 10.1070/RM2006v061n01ABEH004299
- Henry C. Wente, Counterexample to a conjecture of H. Hopf, Pacific J. Math. 121 (1986), no. 1, 193–243. MR 815044, DOI 10.2140/pjm.1986.121.193
- T. J. Willmore, Riemannian geometry, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993. MR 1261641
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
- 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”.
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2813421