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 \(N=\mathbb{S}^{k}\), \(k\geqslant 3\), 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 \(3\leqslant n\leqslant 5\), 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.