Intrinsically biharmonic maps into homogeneous spaces

The tension field $\tau(u)$ of a map $u$ from a domain ­ $\Omega \subset \mathbb{R}^m$ into a manifold $N$ is the negative $L^2$-gradient of the Dirichlet energy. In this paper we study critical points of the intrinsic biharmonic energy functional $T(u) = \int_\Omega |\tau(u)|^2$ when $N$ is a homogeneous space. We derive an Euler-Lagrange equation which makes sense for all critical points of $T$, in contrast to previously known versions. We also obtain a partial regularity result for solutions to this equation for arbitrary domain dimension.