Nonexistence of stable harmonic maps to and from certain homogeneous spaces and submanifolds of Euclidean space

Abstract

Call a compact Riemannian manifold M 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 M contains elements of arbitrarily small energy. If M M is isometrically immersed in Euclidean space, then a condition on the second fundamental form of M M is given which implies M 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.