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Normalized potentials of minimal surfaces in spheres

Published online by Cambridge University Press:  22 January 2016

Quo-Shin Chi ,
Luis Fernández  and
Hongyou Wu
Quo-Shin Chi
Affiliation:
Department of Mathematics, Washington University, St. Louis, MO 63130, U.S.A., [email protected]
Luis Fernández
Affiliation:
Department of Mathematics, Universidad de los Andes, Bogotá, Colombia, [email protected]
Hongyou Wu
Affiliation:
Department of Mathematics, Northern Illinois University, DeKalb, IL 60115, U.S.A., [email protected]
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Abstract

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We determine explicitly the normalized potential, a Weierstrass-type representation, of a superconformal surface in an even-dimensional sphere S2n in terms of certain normal curvatures of the surface. When the Hopf differential is zero the potential embodies a system of first order equations governing the directrix curve of a superminimal surface in the twistor space of the sphere. We construct a birational map from the twistor space of S2n into ℂPn(n+1)/2. In general, birational geometry does not preserve the degree of an algebraic curve. However, we prove that the birational map preserves the degree, up to a factor 2, of the twistor lift of a superminimal surface in S6 as long as the surface does not pass through the north pole. Our approach, which is algebro-geometric in nature, accounts in a rather simple way for the aforementioned first order equations, and as a consequence for the particularly interesting class of superminimal almost complex curves in S6. It also yields, in a constructive way, that a generic superminimal surface in S6 is not almost complex and can achieve, by the above degree property, arbitrarily large area.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 156 , 1999 , pp. 187 - 214
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1999

References

[1] Bolton, J., Pedit, F. and Woodward, L., Minimal surfaces and the Toda field model, J. reine angew. Math., 459 (1995), 119–150.Google Scholar
[2] Bryant, R., Conformal and minimal immersions of compact surfaces into the 4-sphere, J. Differ. Geom., 17 (1982), 455–473.Google Scholar
[3] Bryant, R., Submanifolds and special structures on the octonians, J. Differ. Geom., 17 (1982), 185–232.CrossRefGoogle Scholar
[4] Bryant, R., Lie groups and twistor spaces, Duke Math. J., 52 (1985), 223–261.CrossRefGoogle Scholar
[5] Burstall, F. E., Harmonic tori in spheres and complex projective spaces, J. reine angew. Math., 469 (1995), 149–177.Google Scholar
[6] Calabi, E., Isometric embeddings of complex manifolds, Ann. Math., 58 (1953), 1–23.Google Scholar
[7] Calabi, E., Minimal immersions of surfaces in Euclidean spheres, J. Differ. Geom., 1 (1967), 111–125.Google Scholar
[8] Chern, S. S., On the minimal immersions of the two sphere in a space of constant curvature, Problems in Analysis, Symposium in honor of Solomon Bochner, Princeton Univ. Press, Princeton, 1970, 27–49.Google Scholar
[9] Chi, Q. S. and Mo, X., Rigidity of superminimal surfaces in complex projective spaces, Tôhoku Math. J., 44 (1992), 83–101.Google Scholar
[10] Chi, Q. S. and Mo, X., The moduli space of superminimal surfaces of a fixed degree, genus and conformal structure in the four-sphere, Osaka J. Math, 33 (1996), 669–696.Google Scholar
[11] Dorfmeister, J., McIntosh, I., Pedit, F. and Wu, H., On the meromorphic potential for a harmonic surface in a k-symmetric space, Manuscripta Math., 92 (1997), 143–152.CrossRefGoogle Scholar
[12] Dorfmeister, J., Pedit, F. and Wu, H., Weierstrass type representations of harmonic maps into symmetric spaces, Comm. Anal. and Geom., to appear.Google Scholar
[13] Eells, J. and Lemaire, L., A report on harmonic maps, Bull. London Math. Soc, 10 (1978), 1–68.Google Scholar
[14] Ferus, D., Pedit, F., Pinkall, U. and Sterling, I., Minimal tori in S4 , J. reine angew. Math., 429 (1992), 1–47.Google Scholar
[15] Fernandez, L., Superminimal surfaces in S2n , Thesis, Washington University, 1997.Google Scholar
[16] Guest, M. and Ohnita, Y., Group actions and deformations for harmonic maps, J. Math. Soc. Japan, 45 (1993), 671–704.CrossRefGoogle Scholar
[17] Hano, J., Conformal immersions of compact Riemann surfaces into the 2n-sphere n>2, Nagoya Math. J., 141 (1996), 79–105.Google Scholar
[18] Hitchin, J. N., Harmonic maps from a 2-torus to the 3-sphere, J. Differ. Geom., 31 (1990), 627–710.Google Scholar
[19] Hoffman, D. and Meeks, W., A complete embedded minimal surface with genus one, three ends and finite total curvature, J. Differ. Geom., 21 (1985), 109–127.CrossRefGoogle Scholar
[20] Hoffman, D., Wei, F. S. and Karcher, H., Adding handles to the helicoid, Bull. Amer. Math. Soc, 29 (1993), 77–84.Google Scholar
[21] Kotani, M., Connectedness of the space of minimal 2-spheres in S2m(1), Proc. Amer. Math. Soc, 120 (1994), 803–810.Google Scholar
[22] Karcher, H., Pinkall, U. and Sterling, I., New minimal surfaces in S3 , J. Differ. Geom., 28 (1988), 169–185.Google Scholar
[23] Lawson, B., Complete minimal surfaces in S3 , Ann. Math., 92 (1970), 335–374.Google Scholar
[24] Miyaoka, R., The family of isometric superconformal harmonic maps and the affine Toda equations, J. Reine Angew. Math., 481 (1996), 1–25.Google Scholar
[25] Osserman, R., A Survey of Minimal Surfaces, Dover Publications, Inc., New York, 1986.Google Scholar
[26] Wei, F. S., Some existence and uniqueness theorems for doubly periodic minimal surfaces, Invent. Math., 109 (1992), 113–136.Google Scholar
[27] Wu, H., Banach manifolds of minimal surfaces in the 4-sphere, Amer. Math. Soc. Proc. Symp. Pure Math., 54 (1993), 513–539.Google Scholar
[28] Wu, H., A simple way for determining the normalized potentials for harmonic maps, Ann. Global Anal. Geom., 17 (1999), 189–199.CrossRefGoogle Scholar