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

Abstract

We study the minimization problem for the Yang-Mills energy under fixed boundary connection in supercritical dimension $n\ge 5$. 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-Rivière and Tao-Tian in which respectively a strong approximability property and an admissibility property were assumed in addition.

Proofs were clarified for Proposition 3.1 and Corollary 1.13. Part of the proof of Proposition 2.1 was moved into Theorem B.1