The space of harmonic maps of $S^ 2$ into $S^ 4$
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- by Bonaventure Loo PDF
- Trans. Amer. Math. Soc. 313 (1989), 81-102 Request permission
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
- V. Arnold, Les méthodes mathématiques de la mécanique classique, Éditions Mir, Moscow, 1976 (French). Traduit du russe par Djilali Embarek. MR 0474391
- João Lucas Marquês Barbosa, On minimal immersions of $S^{2}$ into $S^{2m}$, Trans. Amer. Math. Soc. 210 (1975), 75–106. MR 375166, DOI 10.1090/S0002-9947-1975-0375166-2
- 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 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.
- Eugenio Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Differential Geometry 1 (1967), 111–125. MR 233294
- Shiing Shen Chern and Jon Gordon Wolfson, Minimal surfaces by moving frames, Amer. J. Math. 105 (1983), no. 1, 59–83. MR 692106, DOI 10.2307/2374381
- J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), no. 1, 1–68. MR 495450, DOI 10.1112/blms/10.1.1
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
- Paul Gauduchon and H. Blaine Lawson Jr., Topologically nonsingular minimal cones, Indiana Univ. Math. J. 34 (1985), no. 4, 915–927. MR 808834, DOI 10.1512/iumj.1985.34.34050 H. B. Lawson, Surfaces minimales et la construction de Calabi-Penrose, Séminaire Bourbaki, 624 (1984).
- H. Blaine Lawson Jr., Complete minimal surfaces in $S^{3}$, Ann. of Math. (2) 92 (1970), 335–374. MR 270280, DOI 10.2307/1970625
- Claude LeBrun, Spaces of complex null geodesics in complex-Riemannian geometry, Trans. Amer. Math. Soc. 278 (1983), no. 1, 209–231. MR 697071, DOI 10.1090/S0002-9947-1983-0697071-9 B. Loo, Branched superminimal surfaces in ${S^4}$, Ph.D Thesis, State Univ. of New York at Stony Brook, 1987. M. L. Michelsohn, Surfaces minimales dans les sphères, Séminaire de l’Ecole Polytechnique, 1984.
- Graeme Segal, The topology of spaces of rational functions, Acta Math. 143 (1979), no. 1-2, 39–72. MR 533892, DOI 10.1007/BF02392088 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. J. L. Verdier, Applications harmoniques de ${S^2}$ dans ${S^4}$ (preprint). B. L. Van Der Waerden, Algebra, Ungar, New York, 1970.
Additional Information
- © Copyright 1989 American Mathematical Society
- 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