Biharmonic maps into a Riemannian manifold of non-positive curvature (Q2248889)

The authors study biharmonic maps between Riemannian manifolds, where the target manifold in of non-positive curvature. Harmonic maps are critical points of the energy functional \(E(\varphi)=\frac{1}{2}\int_M |d\varphi|^2\text{vol}_M\). The Euler-Lagrange equation is given by the vanishing of the tension field \(\tau(\varphi)\). Biharmonic maps, a natural generalization of harmonic maps, are the critical points of the bienergy functional \(E_2(\varphi)=\frac{1}{2}\int_M|\tau(\varphi)|^2\text{vol}_M\). The authors prove that every biharmonic map \(\varphi:M\to N\) with finite energy and finite bienergy, where \(M\) is complete and \(N\) is of non-positive curvature must be harmonic. The method of the proof implies generalized Chen's conjecture under natural conditions. Namely, it is proved that if \(\varphi:M\to N\) is a biharmonic isometric immersion, \(M\) is complete and \(N\) is of non-positive curvature and the \(L^2\)-norm of the mean curvature of this immersion is finite, then it is minimal.