Compactness in gauge theory - Ewald, Patrícia Muñoz (2021)
In 1982, Karen Uhlenbeck published two foundational papers in gauge theory, which quickly led to Simon Donaldson\'s Fields medal winning result on topology of four-manifolds, and to the beginning of an era of using gauge theoretic techniques as tools for proving theorems. In 2019, she became the first (and as of 2021 only) woman to receive the Abel prize, for these and other groundbreaking works in geometric analysis. In one of these works, entitled \"Connections with L^p bounds on curvature\", Uhlenbeck proved two very important technical results on the existence of a good gauge, and the sequential compactness of weak connections with bounded curvature. In this work, we prove these results and then address their immediate consequence: the uniform convergence of weak Yang-Mills connections with bounded curvature.