Biharmonic maps into compact Lie groups and integrable systems (Q2445032)

The article studies biharmonic maps from compact Riemann manifolds to compact Lie groups and their relation to integrable systems. More precisely, let \((M,g)\) be a compact Riemannian manifold and \((G,h)\) be a compact Lie group with bi-invariant Riemannian metric \(h\). For a map \(\psi:(M,g)\to(G,h)\) the bienergy functional is given by \[ E_2(\psi):=\frac{1}{2}\int_M|\tau(\psi)|^2dv_g, \] where \(\tau\) is the tension field of the map \(\psi\) and critical points of \(E_2\) are called biharmonic maps. After recalling the necessary general material on harmonic maps and biharmonic maps, it is shown that \(\psi\) being biharmonic is equivalent to \[ \delta d\delta\alpha+\text{Trace}_g([\alpha,d\delta\alpha])=0, \] where \(\alpha\) denotes the pullback of the Maurer-Cartan form of \(G\). Using this formula, an existence result for real analytic biharmonic curves into a compact Lie group \((G,h)\) is presented and several examples of such curves are discussed. Moreover, an existence result for biharmonic maps of an open domain of \(\mathbb{R}^2\) with a Riemannian metric conformal to the standard Euclidean metric into \((G,h)\) is established.