Existence of polyharmonic maps in critical dimensions

Abstract

We prove that for any two closed Riemannian manifolds $M^{2m}$ ($m\ge 1$) and $N$, there exists a minimizing (extrinsic) $m$-polyharmonic map for every free homotopy class in $[M^{2m},N]$, provided that the homotopy group ${\pi }_{2m}(N)$ is trivial. This generalizes the celebrated existence results for harmonic maps and biharmonic maps. We also prove that there exists a non-constant smooth polyharmonic map from ${\mathbb{R}}^{2m}$ to $N$ by a blowup analysis at an energy-concentration point for an energy-minimizing sequence if the convergence fails to be strong.

23 pages