×

Normalized potentials of minimal surfaces in spheres. (English) Zbl 0976.53068

Authors’ 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 \(\mathbb{C} P^{n(n+1)/2}\). 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.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C28 Twistor methods in differential geometry
Full Text: DOI

References:

[1] J. Differ. Geom 21 pp 109– (1985) · Zbl 0604.53002 · doi:10.4310/jdg/1214439467
[2] DOI: 10.1090/S0273-0979-1993-00401-4 · Zbl 0787.53003 · doi:10.1090/S0273-0979-1993-00401-4
[3] Nagoya Math. J 141 pp 79– (1996) · Zbl 0856.53046 · doi:10.1017/S0027763000005535
[4] DOI: 10.2969/jmsj/04540671 · Zbl 0792.58012 · doi:10.2969/jmsj/04540671
[5] Thesis (1997) · Zbl 0881.03002
[6] DOI: 10.2748/tmj/1178227378 · Zbl 0794.53039 · doi:10.2748/tmj/1178227378
[7] J. reine angew. Math 429 pp 1– (1992)
[8] Problems in Analysis, Symposium in honor of Solomon Bochner pp 27– (1970)
[9] DOI: 10.1112/blms/10.1.1 · Zbl 0401.58003 · doi:10.1112/blms/10.1.1
[10] J. Differ. Geom 1 pp 111– (1967) · Zbl 0171.20504 · doi:10.4310/jdg/1214427884
[11] Comm. Anal. and Geom
[12] DOI: 10.2307/1969817 · Zbl 0051.13103 · doi:10.2307/1969817
[13] DOI: 10.1007/BF02678186 · Zbl 0903.58005 · doi:10.1007/BF02678186
[14] J. reine angew. Math 469 pp 149– (1995)
[15] Osaka J. Math 33 pp 669– (1996)
[16] DOI: 10.1215/S0012-7094-85-05213-5 · Zbl 0582.58011 · doi:10.1215/S0012-7094-85-05213-5
[17] J. Differ. Geom 17 pp 185– (1982) · Zbl 0526.53055 · doi:10.4310/jdg/1214436919
[18] J. Differ. Geom 17 pp 455– (1982) · Zbl 0498.53046 · doi:10.4310/jdg/1214437137
[19] J. reine angew. Math 459 pp 119– (1995)
[20] DOI: 10.1023/A:1006556302766 · Zbl 0954.58017 · doi:10.1023/A:1006556302766
[21] Amer. Math. Soc. Proc. Symp. Pure Math 54 pp 513– (1993)
[22] DOI: 10.1007/BF01232021 · Zbl 0773.53005 · doi:10.1007/BF01232021
[23] A Survey of Minimal Surfaces (1986)
[24] J. Reine Angew. Math 481 pp 1– (1996)
[25] DOI: 10.2307/1970625 · Zbl 0205.52001 · doi:10.2307/1970625
[26] J. Differ. Geom 28 pp 169– (1988) · Zbl 0653.53004 · doi:10.4310/jdg/1214442276
[27] Proc. Amer. Math. Soc 120 pp 803– (1994)
[28] J. Differ. Geom 31 pp 627– (1990) · Zbl 0725.58010 · doi:10.4310/jdg/1214444631
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.