Minimizers of higher order gauge invariant functionals

Abstract

We introduce higher order variants of the Yang-Mills functional that involve $(n-2)$th order derivatives of the curvature. We prove coercivity and smoothness of critical points in Uhlenbeck gauge in dimensions $\dim M\le 2n$. These results are then used to establish the existence of smooth minimizers on a given principal bundle $P\to M$ for subcritical dimensions $\dim M<2n$. In the case of critical dimension $\dim M=2n$ we construct a minimizer on a bundle which might differ from the prescribed one, but has the same Chern classes $c_1,\dots ,c_{n-1}$. A key result is a removable singularity theorem for bundles carrying a $W^{n-1,2}$-connection. This generalizes a recent result by Petrache and Rivière.