Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

BIMINIMAL IMMERSIONS

E. Loubeaua1 and S. Montaldoa2

a1 Département de Mathématiques, Université de Bretagne, Occidentale 6, avenue Victor Le Gorgeu CS 93837, 29238 Brest Cedex 3, France (loubeau@univ-brest.fr)

a2 Dipartimento di Matematica e Informatica, Università di Cagliari, Via Ospedale 72, 09124 Cagliari, Italy (montaldo@unica.it)

Abstract

We study biminimal immersions: that is, immersions which are critical points of the bienergy for normal variations with fixed energy. We give a geometrical description of the Euler–Lagrange equation associated with biminimal immersions for both biminimal curves in a Riemannian manifold, with particular attention given to the case of curves in a space form, and isometric immersions of codimension 1 in a Riemannian manifold, in particular for surfaces of a three-dimensional manifold. We describe two methods of constructing families of biminimal surfaces using both Riemannian and horizontally homothetic submersions.

(Received April 11 2006)

Keywords

  • Primary 58E20;
  • biminimal surfaces;
  • biharmonic maps;
  • Willmore functional;
  • Thurston geometry

Footnotes

Dedicated to Professor Renzo Caddeo on his 60th birthday.