## Nagoya Mathematical Journal

### Normalized potentials of minimal surfaces in spheres

#### Abstract

We determine explicitly the normalized potential, a Weierstrass-type representation, of a superconformal surface in an even-dimensional sphere $S^{2n}$ 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 $S^{2n}$ into ${\Bbb C}P^{n(n+1)/2}$. 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 $S^{6}$ 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 $S^{6}$. It also yields, in a constructive way, that a generic superminimal surface in $S^{6}$ is not almost complex and can achieve, by the above degree property, arbitrarily large area.

#### Article information

Source
Nagoya Math. J., Volume 156 (1999), 187-214.

Dates
First available in Project Euclid: 27 April 2005

https://projecteuclid.org/euclid.nmj/1114631306

Mathematical Reviews number (MathSciNet)
MR1727900

Zentralblatt MATH identifier
0976.53068

#### Citation

Chi, Quo-Shin; Fernández, Luis; Wu, Hongyou. Normalized potentials of minimal surfaces in spheres. Nagoya Math. J. 156 (1999), 187--214. https://projecteuclid.org/euclid.nmj/1114631306

#### References

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