On superminimal surfaces

Thomas Friedrich

Archivum Mathematicum (1997)

  • Volume: 033, Issue: 1-2, page 41-56
  • ISSN: 0044-8753

Abstract

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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.

How to cite

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Friedrich, Thomas. "On superminimal surfaces." Archivum Mathematicum 033.1-2 (1997): 41-56. <http://eudml.org/doc/248022>.

@article{Friedrich1997,
abstract = {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.},
author = {Friedrich, Thomas},
journal = {Archivum Mathematicum},
keywords = {minimal surfaces; hyperbolic spaces; minimal surfaces; hyperbolic spaces},
language = {eng},
number = {1-2},
pages = {41-56},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On superminimal surfaces},
url = {http://eudml.org/doc/248022},
volume = {033},
year = {1997},
}

TY - JOUR
AU - Friedrich, Thomas
TI - On superminimal surfaces
JO - Archivum Mathematicum
PY - 1997
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 033
IS - 1-2
SP - 41
EP - 56
AB - 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.
LA - eng
KW - minimal surfaces; hyperbolic spaces; minimal surfaces; hyperbolic spaces
UR - http://eudml.org/doc/248022
ER -

References

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  7. Kommerell K., Die Krümmung der zweidimensionalen Gebilde im ebenen Raum von vier Dimensionen, Dissertation Tübingen 1897. 
  8. Kommerell K., Riemannsche Flächen im ebenen Raum von vier Dimensionen, Math. Ann. 60 (1905), 548-596. (1905) MR1511325
  9. Kwietniewski, St., Über Flächen des vierdimensionalen Raumes, deren sämtliche Tangentialebenen untereinander gleichwinklig sind, und ihre Beziehung zu ebenen Kurven, Dissertation Zürich 1902. (1902) 
  10. 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. (1961) 
  11. Schiemangk, Chr., Sulanke R., Submanifolds of the Möbius space, Math. Nachr. 96 (1980), 165-183. (1980) Zbl0484.53008MR0600808

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