Stability of harmonic maps and standard minimal immersions (Q1070600)

By the results of Xin and Leung the standard n-sphere \(S^ n\) with \(n\geq 3\) has the property that (1) a noncontant harmonic map from \(S^ n\) to any Riemannian manifold is unstable, and (2) a nonconstant harmonic map from any compact Riemannian manifold to \(S^ n\) is unstable. If a compact Riemannian manifold has this property, then we call it harmonically unstable. In this note we classify harmonically unstable symmetric spaces. We show that a compact symmetric space M is harmonically unstable, if and only if M is a product of simply connected compact irreducible symmetric spaces belonging to the following list: (i) SU(n) (n\(\geq 2)\), Sp(n) (n\(\geq 2)\), (ii) SU(2n)/Sp(n) (n\(\geq 3)\), (iii) \(S^ n\) (n\(\geq 3)\), (iv) \(Sp(p+q)/Sp(p)\times Sp(q)\) (p\(\geq q\geq 1)\), (v) \(E_ 6/F_ 4\), (vi) \(F_ 4/Spin(9)\). This list is the complete list of compact irreducible symmetric spaces with the unstable identity map as a harmonic map. Furthermore we study the harmonic instability of isoparametric hypersurfaces in a unit sphere.