Nonexistence of Stable Harmonic Maps To and From Certain Homogeneous Spaces and Submanifolds of Euclidean Space

Call a compact Riemannian manifold $M$ a strongly unstable manifold if it is not the range or domain of a nonconstant stable harmonic map and also the homotopy class of any map to or from $M$ contains elements of arbitrarily small energy. If $M$ is isometrically immersed in Euclidean space, then a condition on the second fundamental form of $M$ is given which implies $M$ is strongly unstable. As compact isotropy irreducible homogeneous spaces have "standard" immersions into Euclidean space this allows a complete list of the strongly unstable compact irreducible symmetric spaces to be made.