Counting solutions of perturbed harmonic map equations

Abstract

In this paper we consider perturbed harmonic map equations for maps between closed Riemannian manifolds. In the case where the target manifold has negative sectional curvature we prove - among other results - that for a large class of semilinear and quasilinear perturbations, the perturbed harmonic map equations have solutions in any homotopy class of maps for which the Euler characteristic of the set of harmonic maps does not vanish. Under an additional condition, similar results hold in the case where the target manifold has nonpositive sectional curvature. The proofs are presented in an abstract setup suitable for generalizations to other situations.