Harmonic maps into Grassmann manifolds

Abstract

A harmonic map from a Riemannian manifold into a Grassmann manifold is characterized by a vector bundle, a space of sections of this bundle and a Laplace operator. We apply our main theorem (a generalization of theorem of Takahashi) to generalize the theory of do Carmo and Wallach and to describe the moduli space of harmonic maps satisfying the gauge and the Einstein–Hermitian conditions from a compact Riemannian manifold into a Grassmannian. The geometric meaning of the compactification of the moduli space is interpreted and it is shown that the compactified moduli space is connected and convex. As applications, several rigidity results are exhibited and we also construct moduli spaces of holomorphic isometric embeddings of the complex projective line into complex quadrics of low degree. The compactification of the moduli space leads to classification theorems for equivariant harmonic maps.