Existence of nontrival $n$-harmonic maps via min-max methods

TL;DR

This paper proves the existence of nontrivial n-harmonic maps from spheres into certain manifolds using min-max methods, especially as limits of p-harmonic maps when p approaches n from above.

Contribution

It establishes the existence of nontrivial n-harmonic maps for manifolds with nontrivial higher homotopy groups via min-max techniques, linking them to bubbling limits of p-harmonic maps.

Findings

Existence of nontrivial n-harmonic maps for manifolds with nontrivial higher homotopy groups.

Construction of these maps as bubbling limits of p-harmonic maps as p approaches n.

Application of min-max methods to establish existence results.

Abstract

For any $n \geq 3$ and any closed manifold $\mathcal{N}$ with $\pi_{n+k}(\mathcal{N}) \neq \{0\}$ for some $k \geq 0$, we establish the existence of nontrivial $n$-harmonic maps from $\mathbb{S}^n$ into $\mathcal{N}$. When $k\geq 1$, these maps naturally appear as bubbling limits of $p$-harmonic maps with $p > n$, obtained by min-max constructions in the limit $p \to n^+$.