Uhlenbeck compactness (Q1422443)

The aim of the book is to proof in detail the weak and strong Uhlenbeck compactness (WUC and SUC) theorems [see \textit{K. Uhlenbeck}, Commun. Math. Phys. 83, 11--29 (1982; Zbl 0491.58032) and ibid. 83, 31--42 (1982; Zbl 0499.58019)], and explain their fundamental role in gauge theory. Main results: \(A\)-WUC, a subset of Sobolev space of connections that satisfies an \(L^p\)-bound on the curvature is weakly compact, \(A'\)-generalization for sequence of compact submanifolds, \(B\)-existence of Coulomb type Uhlenbeck gauge transformations, \(C\)-gauge transformations after Agmon, Douglis and Nirenberg, \(D\)-Calderón-Zygmund inequality for the sequence of global gauge transformations, \(E\)-SUC, a sequence of gauge transformations converges uniformly with all derivatives to a smooth connection, \(F\)-local slice theorem, every flat connection is equivalent to a smooth connection, \(F'-L^p\)-local slice theorem, every weakly flat connection is equivalent to a smooth connection. Appendices: \(A\)-introduction to gauge theory, \(B\)-Sobolev spaces of sections of fibre bundles, \(C\)-Mikhlin criteria for \(L^p\)-multipliers, \(D\)-Dirichlet problem, \(E\)-implicit function theorem for several Banach spaces.