A new higher order Yang--Mills--Higgs flow in Riemannian $4$-manifold

TL;DR

This paper introduces a new higher order Yang--Mills--Higgs flow on 4-manifolds, proving long-time existence of solutions under certain conditions and improving previous results for line bundles using novel techniques involving Green functions.

Contribution

It develops a new higher order Yang--Mills--Higgs flow and establishes long-time existence results, including improvements for line bundles with innovative blow-up analysis.

Findings

Solutions avoid finite time singularities under certain conditions

Long-time existence of the flow for line bundles is established

New techniques involving Green functions are introduced

Abstract

Let $(M,g)$ be a closed Riemannian $4$-manifold and let $E$ be a vector bundle over $M$ with structure group $G$, where $G$ is a compact Lie group. In this paper, we consider a new higher order Yang--Mills--Higgs functional, in which the Higgs field is a section of $\Omega^0(\textmd{ad}E)$. We show that, under suitable conditions, solutions to the gradient flow do not hit any finite time singularities. In the case that $E$ is a line bundle, we are able to use a different blow up procedure and obtain an improvement of the long time result in \cite{Z1}. The proof is rather relevant to the properties of the Green function, which is very different from the previous techniques in \cite{Ke,Sa,Z1}.