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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., chi@math.wustl.edu
Luis Fernández
Affiliation:
Department of Mathematics, Universidad de los Andes, Bogotá, Colombia, ifernand@uniandes.edu.co
Hongyou Wu
Affiliation:
Department of Mathematics, Northern Illinois University, DeKalb, IL 60115, U.S.A., wu@math.niu.edu
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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

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