and the corresponding Euler-Lagrange equation is H = 0, where H is the mean curvature vector field. If φ : (M, g) → (N, h) is a Riemannian immersion, then it is a critical point of the bienergy in C∞(M,N) if and only if it is a minimal immersion [26]. Thus, in order to study minimal immersions one can look at harmonic Riemannian immersions. A natural generalization of harmonic maps and minimal ...

In this paper, we consider p-biharmonic submanifolds of aspace form. We give the necessary and sufficient conditions for a submanifold to be in space present some new properties stress p-bienergy tensor.

Biconservative surfaces are with divergence-free stress-bienergy tensor. Simply connected, complete, non-$CMC$ biconservative in $3$-dimensional space forms were constructed working extrinsic and intrinsic ways. Then, one raises the question of uniqueness such surfaces. In this paper we give a positive answer to question.

This paper studies some properties of F-biharmonic maps between Riemannian manifolds. By considering the first variation formula of the F-bienergy functional, F-biharmonicity of conformal maps are investigated. Moreover, the second variation formula for F-biharmonic maps is obtained. As an application, instability and nonexistence theorems for F-biharmonic maps are given.

The main purpose of this paper is to study biharmonic hypersurface in a quasi-paraSasakian manifold $\mathbb{Q}^{2m+1}$. Biharmonic hypersurfaces are special cases maps and the critical points bienergy functional. condition biharmonicity for non-degenerate $\mathbb{Q}^{2m+1}$ investigated both cases: either characteristic vector field unit normal or it belongs tangent space hypersurface. Some r...

in this paper, we study spacelike dual biharmonic curves. we characterize spacelike dual biharmonic curves in terms of their curvature and torsion in the lorentzian dual heisenberg group . we give necessary and sufficient conditions for spacelike dual biharmonic curves in the lorentzian dual heisenberg group . therefore, we prove that all spacelike dual biharmonic curves are spacelike dual heli...

We study biharmonic hypersurfaces in a generic Riemannian manifold. We first derive an invariant equation for such hypersurfaces generalizing the biharmonic hypersurface equation in space forms studied in [16], [8], [6], [7]. We then apply the equation to show that the generalized Chen’s conjecture is true for totally umbilical biharmonic hypersurfaces in an Einstein space, and construct a (2-p...

Abstract Let $B^n\subset \mathbb{R} ^{n}$ and $\mathbb{S} ^n\subset ^{n+1}$ denote the Euclidean $n$-dimensional unit ball sphere, respectively. The extrinsic $k$-energy functional is defined on Sobolev space $W^{k,2}\left (B^n,\mathbb{S} ^n \right )$ as follows: $E_{k}^{{\textrm{ext}}}(u)=\int _{B^n}|\Delta ^s u|^2\ dx$ when $k=2s$, _{B^n}|\nabla \Delta $k=2s+1$. These energy functionals are a...