Existence of harmonic maps and eigenvalue optimization in higher dimensions

Abstract We prove the existence of nonconstant harmonic maps of optimal regularity from an arbitrary closed manifold (M^n, g) (M n, g) of dimension n>2 n > 2 to any closed, non-aspherical manifold N N containing no stable minimal two-spheres. In particular, this gives the first general existence result for harmonic maps from higher-dimensional manifolds to a large class of positively curved targets. In the special case of the round spheres N=S^k N = S k, k 3 k ⩾ 3, we obtain a distinguished family of nonconstant harmonic maps M S^k M → S k of index at most k+1 k + 1, with singular set of codimension at least 7 for k k sufficiently large. Furthermore, if 3 n 5 3 ⩽ n ⩽ 5, we show that these smooth harmonic maps stabilize as k k becomes large, and correspond to the solutions of an eigenvalue optimization problem on M M, generalizing the conformal maximization of the first Laplace eigenvalue on surfaces.