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Some examples of critical points for the total mean curvature functional

Published online by Cambridge University Press:  20 January 2009

Josu Arroyo ,
Manuel Barros  and
Oscar J. Garay
Josu Arroyo
Affiliation:
Departamento de Matemáticas, Universidad del País Vasco/Euskal, Herriko Unibertsitatea, Aptdo 644, 48080 Bilbao, Spain (mtparolj@lg.ehu.es; mtpgabeo@lg.ehu.es)
Manuel Barros
Affiliation:
Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain (mbarros@goliat.ugr.es)
Oscar J. Garay
Affiliation:
Departamento de Matemáticas, Universidad del País Vasco/Euskal, Herriko Unibertsitatea, Aptdo 644, 48080 Bilbao, Spain (mtparolj@lg.ehu.es; mtpgabeo@lg.ehu.es)
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Abstract

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We study the following problem: establish existence and classification of closed curves which are critical points for the total curvature functional, defined on spaces of curves in a Riemannian manifold. This problem is completely solved in a real space form. Next, we give examples of critical points for this functional in a class of metrics with constant scalar curvature on the three sphere. Also, we obtain a rational one-parameter family of closed helices which are critical points for that functional in ℂℙ2 (4) when it is endowed with its usual Kaehlerian structure. Finally, we use the principle of symmetric criticality to get equivariant submanifolds, constructed on the above curves, which are critical points for the total mean curvature functional.

Keywords

total curvaturecritical pointstotal tension
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 43 , Issue 3 , October 2000 , pp. 587 - 603
Copyright
Copyright © Edinburgh Mathematical Society 2000

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