Biminimal Immersions

Abstract

Dedicated to Professor Renzo Caddeo on his 60th birthday.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.

Affine biharmonic submanifolds in 3-dimensional pseudo-Hermitian geometry

“…In [28], the second named author showed that a Hopf cylinder in a Sasakian space form M 3 (H ) is biminimal if and only if its base curve is (H + 3)-biminimal. Note that the S 3 -case was proved in [32].…”

$f$-Biminimal immersions

“…An immersion φ is called biminimal [12] if it is a critical point of the bienergy functional E 2 (φ) for variations normal to the image φ(M ) ⊂ N , with fixed energy. Equivalently, there exists a constant λ ∈ R such that φ is a critical point of the λ -bienergy…”

“…Then it is a free biminimal surface [12], where × t 2 denotes the warped product. Hence, from Theorem 4.1,…”

Biharmonic hypersurfaces with bounded mean curvature

“…Biminimal hypersurfaces were introduced by E. Loubeau and S. Montaldo (cf. [10]). We call an biminimal hypersurface free biminimal if it satisfies the biminimal condition for λ = 0.…”