Removable singularities of stationary fields. (English) Zbl 0814.58015

Rassias, Themistocles M. (ed.), The problem of Plateau. A tribute to Jesse Douglas and Tibor Radó. London: World Scientific Publishing Co. Pte. Ltd. 190-220 (1992).
Let \(B^ n\) be the unit ball in \(\mathbb{R}^ n\). Denote a Yang-Mills field by the 2-form \(F_ A\), where \(A\) is a connection 1-form on a vector bundle \(X\). It is known that in dimensions 2 and 3, all finite energy Yang-Mills fields are regular; in dimensions 5 and higher, singular fields with finite energy can be constructed. It is natural to ask whether singularities in dimensions \(n > 4\) can be removed. In this paper, the author proves that this is true under some conditions. At first, the author shows (Theorem 1.5): Let \(F_ A\) be a \(C^ \infty\) solution of the Yang-Mills equations in \(B^ n/\Sigma\), where \(n> 4\) and \(\Sigma\) is a smooth compact submanifold of codimension \(k\) \((k > 4)\). If \(F_ A \in L^{n/2} (B^ n)\), then \(F_ A\) is equivalent via a continuous gauge transformation to a solution \(F_{\widetilde{A}} \in C^ \infty (B^ n)\). By using this result, the author investigates the removability of isolated and accumulation-point singularities from low energy gauge fields. The field equations studied by physicists generally involve coupled fields: an externally imposed gauge field is assumed to interact with the field induced by a particle. In section 3, the author extends his results to Yang-Mills fields coupled to matter fields. After that, removability of singularities in harmonic maps is discussed. Finally, an a priori inequality for a relevant class of elliptic subsolutions is proved.
For the entire collection see [Zbl 0780.00047].


58E20 Harmonic maps, etc.
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals