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 $\varphi :R^{4}\to S^{4}$ ^[1]
The inverse of the hyperbolic stereographic projection is a proper biharmonic $\varphi :B^{4}\to H^{4}$ ^[2]
The axially symmetric diffeomorphisms of $R^{m}\setminus \{0\}$ , $\varphi (x)={\frac {x}{|x|^{\ell }}}$ , is a proper biharmonic map if and only if $m=\ell +2$ .^[3]^[4] For $\ell =2$ this example is the well known Kelvin transformation.

References

↑ Baird, Paul. "Conformal and semi-conformal biharmonic maps". Annals of Global Analysis and Geometry. 34: 403–414. doi:10.1007/s10455-008-9118-8.
↑ "Biharmonic maps and morphisms from conformal mappings. Tohoku Math. J. 62 (2010), 55–73".
↑ Baird, Paul. "On constructing biharmonic maps and metrics". Annals of Global Analysis and Geometry. 23: 65–75. doi:10.1023/A:1021213930520.
↑ Balmuş, A. "Biharmonic maps between warped product manifolds". Journal of Geometry and Physics. 57: 449–466. doi:10.1016/j.geomphys.2006.03.012.

External links

A short survey on biharmonic maps between Riemannian manifolds

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[1] Baird, Paul. "Conformal and semi-conformal biharmonic maps". Annals of Global Analysis and Geometry. 34: 403–414. doi:10.1007/s10455-008-9118-8.

[2] "Biharmonic maps and morphisms from conformal mappings. Tohoku Math. J. 62 (2010), 55–73".

[3] Baird, Paul. "On constructing biharmonic maps and metrics". Annals of Global Analysis and Geometry. 23: 65–75. doi:10.1023/A:1021213930520.

[4] Balmuş, A. "Biharmonic maps between warped product manifolds". Journal of Geometry and Physics. 57: 449–466. doi:10.1016/j.geomphys.2006.03.012.