Geodesic
On superminimal surfaces
Archivum mathematicum, Tome 33 (1997) no. 1-2, pp. 41-56
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 53A35, 53C42, 58E20
Keywords: minimal surfaces; hyperbolic spaces 
@article{ARM_1997_33_1-2_a6,
     author = {Friedrich, Thomas},
     title = {On superminimal surfaces},
     journal = {Archivum mathematicum},
     pages = {41--56},
     year = {1997},
     volume = {33},
     number = {1-2},
     mrnumber = {1464300},
     zbl = {1022.53050},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_1997_33_1-2_a6/}
}
TY  - JOUR
AU  - Friedrich, Thomas
TI  - On superminimal surfaces
JO  - Archivum mathematicum
PY  - 1997
SP  - 41
EP  - 56
VL  - 33
IS  - 1-2
UR  - http://geodesic.mathdoc.fr/item/ARM_1997_33_1-2_a6/
LA  - en
ID  - ARM_1997_33_1-2_a6
ER  - 
%0 Journal Article
%A Friedrich, Thomas
%T On superminimal surfaces
%J Archivum mathematicum
%D 1997
%P 41-56
%V 33
%N 1-2
%U http://geodesic.mathdoc.fr/item/ARM_1997_33_1-2_a6/
%G en
%F ARM_1997_33_1-2_a6
Friedrich, Thomas. On superminimal surfaces. Archivum mathematicum, Tome 33 (1997) no. 1-2, pp. 41-56. http://geodesic.mathdoc.fr/item/ARM_1997_33_1-2_a6/

[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

[Br] Bryant R. L.: Conformal and minimal immersions of compact surfaces into the 4-sphere. Journ. Diff. Geom. 17 (1982), 455-473. | MR | Zbl

[E] Eisenhart L. P.: Minimal surfaces in Euclidean four-spaces. Amer. Journ. of Math. 34 (1912), 215-236. | MR

[F] Friedrich, Th.: On Surfaces in four-spaces. Ann. Glob. Anal. and Geom. No. 3 (1984), 257-287. | MR | Zbl

[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

[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 | Zbl