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 govern...
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Cambridge University Press (CUP)
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Online Access: | http://dx.doi.org/10.1017/s0027763000007133 https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0027763000007133 |
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crcambridgeupr:10.1017/s0027763000007133 2024-03-03T08:47:15+00:00 Normalized potentials of minimal surfaces in spheres Chi, Quo-Shin Fernández, Luis Wu, Hongyou 1999 http://dx.doi.org/10.1017/s0027763000007133 https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0027763000007133 en eng Cambridge University Press (CUP) https://www.cambridge.org/core/terms Nagoya Mathematical Journal volume 156, page 187-214 ISSN 0027-7630 2152-6842 General Mathematics journal-article 1999 crcambridgeupr https://doi.org/10.1017/s0027763000007133 2024-02-08T08:30:58Z 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 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 in Journal/Newspaper North Pole Cambridge University Press North Pole Nagoya Mathematical Journal 156 187 214 |
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Cambridge University Press |
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English |
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General Mathematics |
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General Mathematics Chi, Quo-Shin Fernández, Luis Wu, Hongyou Normalized potentials of minimal surfaces in spheres |
topic_facet |
General Mathematics |
description |
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 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. |
format |
Article in Journal/Newspaper |
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 |
Cambridge University Press (CUP) |
publishDate |
1999 |
url |
http://dx.doi.org/10.1017/s0027763000007133 https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0027763000007133 |
geographic |
North Pole |
geographic_facet |
North Pole |
genre |
North Pole |
genre_facet |
North Pole |
op_source |
Nagoya Mathematical Journal volume 156, page 187-214 ISSN 0027-7630 2152-6842 |
op_rights |
https://www.cambridge.org/core/terms |
op_doi |
https://doi.org/10.1017/s0027763000007133 |
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Nagoya Mathematical Journal |
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156 |
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187 |
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214 |
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1792503408929800192 |