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