Almost minimizing Yang$-$Mills fields: log-epiperimetric inequality, non-concentration, and uniqueness of tangents

TL;DR

This paper introduces a log-epiperimetric inequality for Yang-Mills fields, proving tangent cone uniqueness at isolated singularities and excluding curvature concentration, advancing understanding of singularities in gauge theory.

Contribution

It develops a novel log-epiperimetric inequality for Yang-Mills fields and proves tangent cone uniqueness at isolated singularities under certain regularity conditions.

Findings

Established a log-epiperimetric inequality for Yang-Mills fields.

Proved uniqueness of tangent cones at isolated singularities.

Excluded curvature concentration along blow-up sequences.

Abstract

We establish a direct log-epiperimetric inequality for Yang$-$Mills fields in arbitrary dimension and we leverage on it to prove uniqueness of tangent cones with isolated singularity for energy minimizing Yang$-$Mills fields and $\omega$-ASD connections (where $\omega$ is not necessarily closed) satisfying some suitable regularity assumptions. En route to this we establish a Luckhaus type lemma for Yang$-$Mills connections to exclude curvature concentration along blow-up sequences.