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)

 
 

 

The space of harmonic maps of $ S\sp 2$ into $ S\sp 4$


Author: Bonaventure Loo
Journal: Trans. Amer. Math. Soc. 313 (1989), 81-102
MSC: Primary 58E20; Secondary 58D15
DOI: https://doi.org/10.1090/S0002-9947-1989-0962283-9
MathSciNet review: 962283
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Every branched superminimal surface of area $ 4\pi d$ in $ {S^4}$ is shown to arise from a pair of meromorphic functions $ ({f_1},{f_2})$ of bidegree $ (d,d)$ such that $ {f_1}$ and $ {f_2}$ have the same ramification divisor. Conditions under which branched superminimal surfaces can be generated from such pairs of functions are derived. For each $ d \geq 1$ the space of harmonic maps (i.e branched superminimal immersions) of $ {S^2}$ into $ {S^4}$ of harmonic degree $ d$ is shown to be a connected space of complex dimension $ 2d + 4$ .


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

  • [1] V. I. Arnold, Mathematical methods of classical mechanics, Springer-Verlag, New York, 1978. MR 0690288 (57:14033b)
  • [2] L. Barbosa, On minimal immersions of $ {S^2}$ into $ {S^{2m}}$ , Trans. Amer. Math. Soc. 210 (1975), 75-106. MR 0375166 (51:11362)
  • [3] R. Bryant, Conformal and minimal immersions of compact surfaces into the $ 4$-spheres, J. Differential Geometry 17 (1982), 455-473. MR 679067 (84a:53062)
  • [4] E. Calabi, Quelques applications de l'analyse complexe aux surfaces d'aire minima, Topics in Complex Manifolds (Ed., H. Rossi), Les Presses de l'Univ. de Montréal, 1967, pp. 59-81.
  • [5] E. Calabi, Minimal immersions of surfaces in euclidean spheres, J. Differential Geometry 1 (1967), 111-125. MR 0233294 (38:1616)
  • [6] S. S. Chern and J. G. Wolfson, Minimal surfaces by moving frames, Amer. J. Math. 105 (1983), 59-83. MR 692106 (84i:53056)
  • [7] J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1-68. MR 495450 (82b:58033)
  • [8] P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley-Interscience, New York, 1978. MR 507725 (80b:14001)
  • [9] P. Gauduchon and H. B. Lawson, Topologically nonsingular minimal cones, Indiana Univ. Math. J. 34 (1985), 915-927. MR 808834 (87h:53083)
  • [10] H. B. Lawson, Surfaces minimales et la construction de Calabi-Penrose, Séminaire Bourbaki, 624 (1984).
  • [11] H. B. Lawson, Complete minimal surfaces in $ {S^3}$, Ann. of Math. 92 (1970), 335-374. MR 0270280 (42:5170)
  • [12] C. LeBrun, Spaces of complex null geodesies in complex-riemannian geometry, Trans. Amer. Math. Soc. 278 (1983), 209-231. MR 697071 (84e:32023)
  • [13] B. Loo, Branched superminimal surfaces in $ {S^4}$, Ph.D Thesis, State Univ. of New York at Stony Brook, 1987.
  • [14] M. L. Michelsohn, Surfaces minimales dans les sphères, Séminaire de l'Ecole Polytechnique, 1984.
  • [15] G. Segal, The topology of spaces of rational functions, Acta Math. 143 (1979), 39-72. MR 533892 (81c:55013)
  • [16] J. L. Verdier, Two dimensional $ \sigma $-models and harmonic maps from $ {S^2}$ to $ {S^{2n}}$, Lecture Notes in Physics, vol. 180, Springer, 1982, pp. 136-141.
  • [17] J. L. Verdier, Applications harmoniques de $ {S^2}$ dans $ {S^4}$ (preprint).
  • [18] B. L. Van Der Waerden, Algebra, Ungar, New York, 1970.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58E20, 58D15

Retrieve articles in all journals with MSC: 58E20, 58D15


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0962283-9
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society