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Uhlenbeck compactness. (English) Zbl 1055.53027

EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society (ISBN 3-03719-004-3/pbk). vii, 212 p. (2004).
The aim of the book is to proof in detail the weak and strong Uhlenbeck compactness (WUC and SUC) theorems [see 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.

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
46E99 Linear function spaces and their duals
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
53C05 Connections (general theory)
58J32 Boundary value problems on manifolds
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