Existence of harmonic maps and eigenvalue optimization in higher dimensions

We prove the existence of nonconstant harmonic maps of optimal regularity from an arbitrary closed manifold (M^n,g) of dimension n>2 to any closed, non-aspherical manifold 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 , k⩾ 3 , we obtain a distinguished family of nonconstant harmonic maps M→𝕊^k of index at most k+1 , with singular set of codimension at least 7 for k sufficiently large. Furthermore, if 3⩽ n⩽ 5 , we show that these smooth harmonic maps stabilize as k becomes large, and correspond to the solutions of an eigenvalue optimization problem on M , generalizing the conformal maximization of the first Laplace eigenvalue on surfaces.