Monotonicity for p-harmonic vector bundle-valued k-forms

TL;DR

This paper studies the monotonicity properties of p-harmonic vector bundle-valued forms, unifies proofs for related monotonicity formulas, and derives Liouville-type theorems, advancing understanding in geometric analysis and gauge theory.

Contribution

It provides a unified approach to monotonicity formulas for p-harmonic maps, Yang-Mills, and Yang-Mills-Higgs pairs, and introduces new Liouville theorems for these objects.

Findings

Unified proof of monotonicity formulas for p-harmonic maps and Yang-Mills connections.

Derived a monotonicity formula for p-Yang-Mills connections.

Established Liouville-type theorems for p-harmonic forms and Yang-Mills-Higgs pairs.

Abstract

We investigate monotonicity properties of $p$-harmonic vector bundle-valued $k$-forms by studying the energy-momentum tensor associated with such a form. As a consequence, we obtain a unified proof of the monotonicity formul{\ae} for $p$-harmonic maps and Yang-Mills connections, proving a monotonicity formula for $p$-Yang-Mills connections in the process. Moreover, it is shown how this technique may be adapted to yield an analogous monotonicity formula for Yang-Mills-Higgs pairs. Finally, we obtain Liouville-type theorems for such forms and Yang-Mills-Higgs pairs as an application.