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 $N={\mathbb{S}}^k$, $k\ge 3$, we obtain a distinguished family of nonconstant harmonic maps $M\to {\mathbb{S}}^k$ of index at most $k+1$, with singular set of codimension at least $7$ for $k$ sufficiently large. Furthermore, if $3\le n\le 5$, 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.