Publication date:
20 August 2017
Source:Advances in Mathematics, Volume 316
Author(s): Mircea Petrache, Tristan Rivière
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 ${\mathcal{A}}_G$ in which to formulate this problem in analogy to the space of integral currents used for the classical Plateau problem. The space ${\mathcal{A}}_G$ 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 existence of weak solutions to the Yang–Mills Plateau problem in the space ${\mathcal{A}}_G$. 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 $L^2$ curvatures previous works of Meyer–Rivière and Tao–Tian in which respectively a strong approximability property and an admissibility property were assumed in addition.