The resolution of the Yang-Mills Plateau problem in super-critical dimensions

TL;DR

This paper addresses the existence, approximation, and regularity of weak solutions to the Yang-Mills Plateau problem in super-critical dimensions, extending previous theories to a broader weak connection framework.

Contribution

It introduces a new function space for weak connections, proves existence and approximation theorems, and establishes optimal regularity for Yang-Mills minimizers in super-critical dimensions.

Findings

Existence of energy-minimizing weak connections in the new space A_G.

Weak connections can be approximated by classical connections with defects.

Proved optimal regularity for Yang-Mills local minimizers.

Abstract

We study the minimization problem for the Yang-Mills energy under fixed boundary connection in supercritical dimension $n\geq 5$. We define the natural function space A_{G} in which to formulate this problem in analogy to the space of integral currents used for the classical Plateau problem. The space A_{G} can be also interpreted as a space of weak connections on a "real measure theoretic version" of reflexive sheaves from complex geometry. We prove the weak closure result which ensures the existence of energy-minimizing weak connections in A_{G}. We then prove that any weak connection from A_{G} can be obtained as a L^2-limit of classical connections over bundles with defects. This approximation result is then extended to a Morrey analogue. We prove the optimal regularity result for Yang-Mills local minimizers. On the way to prove this result we establish a Coulomb gauge extraction theorem for weak curvatures with small Yang-Mills density. This generalizes to the general framework of weak $L^2$ curvatures previous works of Meyer-Rivi\`ere and Tao-Tian in which respectively a strong approximability property and an admissibility property were assumed in addition.