Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
minimal surfaces; hyperbolic spaces
minimal surfaces; hyperbolic spaces
Summary:
Using the Cartan method O. Boruvka (see [B1], [B2]) studied superminimal surfaces in four-dimensional space forms. In particular, he described locally the family of all superminimal surfaces and classified all of them with a constant radius of the indicatrix. We discuss the mentioned results from the point of view of the twistor theory, providing some new proofs. It turns out that the superminimal surfaces investigated by geometers at the beginning of this century as well as by O. Boruvka have a holomorphic and horizontal lift into the twistor space. Global results concerning superminimal surfaces have been obtained during the last 15 years. In this paper we investigate superminimal surfaces in the hyperbolic four-spaces.
References:
[B1] Borůvka O.: Sur une classe de surfaces minima plone’es dans un espace á quatre dimensions a courbure constante. C.R. Acad. Sci 187 (1928), 334-336.
[B2] Borůvka O.: Sur une classe de surfaces minima plouge’es un espace á cinq dimension á courbure constante. C.R. Acad. Sci. 187 (1928), 1271-1273.
[BFG] Beals M., Fefferman C., and Grossman R.: Strictly pseudoconvex domains in $\mathbb C^n$. Bull. Amer. Math. Soc., vol. 8 (1983), 125-326. MR 0684898
[Br] Bryant R. L.: Conformal and minimal immersions of compact surfaces into the 4-sphere. Journ. Diff. Geom. 17 (1982), 455-473. MR 0679067 | Zbl 0498.53046
[E] Eisenhart L. P.: Minimal surfaces in Euclidean four-spaces. Amer. Journ. of Math. 34 (1912), 215-236. MR 1506152
[F] Friedrich, Th.: On Surfaces in four-spaces. Ann. Glob. Anal. and Geom. No. 3 (1984), 257-287. MR 0777909 | Zbl 0562.53039
[K.1] Kommerell K.: Die Krümmung der zweidimensionalen Gebilde im ebenen Raum von vier Dimensionen. Dissertation Tübingen 1897.
[K.2] Kommerell K.: Riemannsche Flächen im ebenen Raum von vier Dimensionen. Math. Ann. 60 (1905), 548-596. MR 1511325
[Kw] Kwietniewski, St.: Über Flächen des vierdimensionalen Raumes, deren sämtliche Tangentialebenen untereinander gleichwinklig sind, und ihre Beziehung zu ebenen Kurven. Dissertation Zürich 1902.
[L] Lumiste Ü.: On the theory of two-dimensional minimal surfaces I-IV. Tartu Riikl. Ül. Tomestised vol. 102 (1961), 3-15 and 16-28, vol. 129 (1962), 74-89 and 90-102.
[S] Schiemangk, Chr., Sulanke R.: Submanifolds of the Möbius space. Math. Nachr. 96 (1980), 165-183. MR 0600808 | Zbl 0484.53008
Search
Partner of
