An energy gap for Yang-Mills connections

TL;DR

This paper establishes a lower bound on the energy of Yang-Mills connections over certain Riemannian manifolds, showing that non-flat connections must have energy exceeding a positive threshold.

Contribution

It proves an energy gap theorem for Yang-Mills connections on manifolds satisfying specific geometric conditions, extending understanding of the energy landscape.

Findings

Energy of non-flat Yang-Mills connections is bounded below by a positive constant.

Flat connections are the only ones with zero energy under the given conditions.

The result applies to both compact and complete Riemannian manifolds.

Abstract

Consider a Yang-Mills connection over a Riemann manifold $M=M^n$, $n\ge 3$, where $M$ may be compact or complete. Then its energy must be bounded from below by some positive constant, if $M$ satisfies certain conditions, unless the connection is flat.