Willmore submanifolds in a Riemannian manifold

Let Nn+p be an (n + p)-dimensional Riemannian manifold and x : M -> Nn+p an isometric immersion of an n-dimensional Riemannian manifold M. A mapping x : M -> Nn+p is called Willmore if it is an extremal submanifold of the following Willmore functional: W(x) = integral(M) (S - nH(2)) (n/2) dv, where S = Sigma(alpha,i,j)(h(ij)(alpha))(2) and H are respectively the norm square of the second fundamental form and the mean curvature of the immersion x, dv is the volume element of M. In this survey paper, we calculate the Euler-Lagrangian equation of W(x) for an n-dimensional submanifold in an (n + p)-dimensional Riemannian manifold Nn+p and give many applications as well as many examples of Willmore submanifolds.