Existence of harmonic maps and eigenvalue optimization in higher dimensions

Abstract

AbstractWe 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 = S k , $k\geqslant 3$ k ⩾ 3 , we obtain a distinguished family of nonconstant harmonic maps $M\to \mathbb{S}^{k}$ 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 $3\leqslant n\leqslant 5$ 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.