Biharmonic map

In mathematics, a biharmonic map is a (smooth) map $\varphi :(M^{m},g)\to (N^{n},h)$ between Riemannian manifolds which is a critical point of the bienergy functional

$E_{2}(\varphi )={\frac {1}{2}}\,\int _{M}|\tau (\varphi )|^{2}\,dv_{g},$

where $\tau (\varphi )\ {\stackrel {\text{def}}{=}}\ \operatorname {trace} _{g}\nabla \,d\varphi$ is the tension field of the map and $dv_{g}$ denotes the volume measure on $M$ induced by its metric. Harmonic maps are characterised by the vanishing of their tension field, thus they are trivially biharmonic. For this reason, the biharmonic maps which are not harmonic are called proper biharmonic.

Examples

• The inverse of the spherical stereographic projection is a proper biharmonic $\varphi :R^{4}\to S^{4}$
• The inverse of the hyperbolic stereographic projection is a proper biharmonic $\varphi :B^{4}\to H^{4}$
• The axially symmetric diffeomorphisms of $R^{m}\setminus \{0\}$, $\varphi (x)={\frac {x}{|x|^{\ell }}}$, is a proper biharmonic map if and only if $m=\ell +2$. For $\ell =2$ this example is the well known Kelvin transformation.

External links

• A short survey on biharmonic maps between Riemannian manifolds