The main topic of the book is a careful and detailed exposition of the two main Uhlenbeck theorems for gauge fields together with their full proofs. Necessary prerequisities are prepared in the first part of the book (regularity and existence problems for L2 and Lp spaces for the Neumann problem) and in five appendices (notation for gauge theories, basic estimates, the Sobolev spaces of sections of vector bundles, the Sobolev embedding and trace theorems, Lp –multipliers, Poisson kernels, basic functional analysis). The two main chapters contain full proofs of the weak and strong compactness theorems. The statements are generalized from closed manifolds to manifolds with boundaries. The proof of the weak compactness theorem follows the original proof of Uhlenbeck. The proof of the strong compactness theorem uses an alternative approach by Salamon, a lot of attention is paid to the local gauge theorem and patching constructions. The book is surely a useful addition to the literature.

Reviewer:

vs