Existence of harmonic maps and eigenvalue optimization in higher dimensions

TL;DR

This paper proves the existence of nonconstant harmonic maps from higher-dimensional manifolds to certain positively curved targets, linking harmonic map existence to eigenvalue optimization in higher dimensions.

Contribution

It establishes the first general existence results for harmonic maps from higher-dimensional manifolds to a broad class of positively curved targets, and connects these maps to eigenvalue optimization problems.

Findings

Existence of harmonic maps from manifolds of dimension > 2 to non-aspherical targets.

Construction of harmonic maps to spheres with bounded index and controlled singular set.

Harmonic maps stabilize and relate to eigenvalue optimization for dimensions 3 to 5.

Abstract

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 $N=\mathbb{S}^k$, $k\geq 3$, 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 $3\leq n\leq 5$, 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.