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≥5. We define the natural function space AG in which to formulate this problem in analogy to the space of integral currents used for the classical Plateau problem. The space AG 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 existence of weak solutions to the Yang–Mills Plateau problem in the space AG.We then prove the optimal regularity result for solutions of this Plateau problem. 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 L2 curvatures previous works of Meyer–Rivière and Tao–Tian in which respectively a strong approximability property and an admissibility property were assumed in addition.