Normalized potentials of minimal surfaces in spheres
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 go...
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ftculeuclid:oai:CULeuclid:euclid.nmj/1114631306 2023-05-15T17:39:54+02:00 Normalized potentials of minimal surfaces in spheres Chi, Quo-Shin Fernández, Luis Wu, Hongyou 1999 application/pdf http://projecteuclid.org/euclid.nmj/1114631306 en eng Duke University Press 0027-7630 Copyright 1999 Editorial Board, Nagoya Mathematical Journal 53C43 58E20 Text 1999 ftculeuclid 2018-10-06T12:38:11Z 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 {\em not} almost complex and can achieve, by the above degree property, arbitrarily large area. Text North Pole Project Euclid (Cornell University Library) North Pole |
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53C43 58E20 Chi, Quo-Shin Fernández, Luis Wu, Hongyou Normalized potentials of minimal surfaces in spheres |
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53C43 58E20 |
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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 {\em not} almost complex and can achieve, by the above degree property, arbitrarily large area. |
format |
Text |
author |
Chi, Quo-Shin Fernández, Luis Wu, Hongyou |
author_facet |
Chi, Quo-Shin Fernández, Luis Wu, Hongyou |
author_sort |
Chi, Quo-Shin |
title |
Normalized potentials of minimal surfaces in spheres |
title_short |
Normalized potentials of minimal surfaces in spheres |
title_full |
Normalized potentials of minimal surfaces in spheres |
title_fullStr |
Normalized potentials of minimal surfaces in spheres |
title_full_unstemmed |
Normalized potentials of minimal surfaces in spheres |
title_sort |
normalized potentials of minimal surfaces in spheres |
publisher |
Duke University Press |
publishDate |
1999 |
url |
http://projecteuclid.org/euclid.nmj/1114631306 |
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North Pole |
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North Pole |
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North Pole |
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North Pole |
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0027-7630 |
op_rights |
Copyright 1999 Editorial Board, Nagoya Mathematical Journal |
_version_ |
1766140668124594176 |