On minimal immersions of 𝑆² into 𝑆^{2𝑚}

Barbosa, Jo ao Lucas Marquês

doi:10.1090/S0002-9947-1975-0375166-2

On minimal immersions of $S^{2}$ into $S^{2m}$
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Trans. Amer. Math. Soc. 210 (1975), 75-106 Request permission

Abstract:

The study of minimal immersions of the $2$-sphere into the standard $n$-sphere of the euclidean space has been better accomplished by associating to each such immersion a certain holomorphic curve. This has been done in several ways in the literature. In the present paper we explore this technique applying some knowledge about the topological and analytical invariants of the particular set of holomorphic curves used to obtain further results. Some new examples are provided, a beginning of a general description of such immersions is given and a rigidity theorem is proved.

References

Eugenio Calabi, Isometric imbedding of complex manifolds, Ann. of Math. (2) 58 (1953), 1–23. MR 57000, DOI 10.2307/1969817
Eugenio Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Differential Geometry 1 (1967), 111–125. MR 233294

Quelques applications de l’analyse complexe aux surfaces d’aire minima

Shiing Shen Chern, On the minimal immersions of the two-sphere in a space of constant curvature, Problems in analysis (Lectures at the Sympos. in honor of Salomon Bochner, Princeton Univ., Princeton, N.J., 1969) Princeton Univ. Press, Princeton, N.J., 1970, pp. 27–40. MR 0362151
Shiing-shen Chern, On minimal spheres in the four-sphere, Studies and Essays (Presented to Yu-why Chen on his 60th Birthday, April 1, 1970) Nat. Taiwan Univ., Math. Res. Center, Taipei, 1970, pp. 137–150. MR 0278205
S. S. Chern, Complex manifolds without potential theory, Van Nostrand Mathematical Studies, No. 15, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0225346
S. S. Chern, Minimal submanifolds in a Riemannian manifold, University of Kansas, Department of Mathematics Technical Report 19 (New Series), University of Kansas, Lawrence, Kan., 1968. MR 0248648
Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387
W. V. D. Hodge and D. Pedoe, Methods of algebraic geometry. Vol. I, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1994. Book I: Algebraic preliminaries; Book II: Projective space; Reprint of the 1947 original. MR 1288305, DOI 10.1017/CBO9780511623899
Hung-hsi Wu, The equidistribution theory of holomorphic curves, Annals of Mathematics Studies, No. 64, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970. MR 0273070