Local monotonicity for the Yang-Mills-Higgs flow

TL;DR

This paper establishes a local monotonicity formula for the Yang-Mills-Higgs flow on G-bundles over Euclidean space, linking it to known non-local formulas and enhancing understanding of flow behavior.

Contribution

It introduces a new local monotonicity formula for the Yang-Mills-Higgs flow, connecting it with existing non-local formulas for better analysis of the flow.

Findings

Proves a local monotonicity formula for the flow.

Shows the monotone quantity matches on self-similar solutions.

Connects local and non-local monotonicity formulas.

Abstract

A local monotonicity formula for the Yang-Mills-Higgs flow on $G$-bundles over $\mathbb{R}^{n}$ ($n>4$) is proved. It is shown that the monotone quantity co\"incides on certain self-similar solutions with that appearing in existing non-local monotonicity formul\ae\ for the Yang-Mills and Yang-Mills-Higgs flows.