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 (Mⁿ,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=Sᵏ N = S k , k 3 k ⩾ 3 , we obtain a distinguished family of nonconstant harmonic maps M→ Sᵏ 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.