Unique continuation theorems in gauge theories
This paper proves five unique continuation theorems for solutions of the anti-self dual equations on 4-dimensional Riemannian manifolds. The first is an analog of \textit{N. Aronszajn's} unique continuation principle [J. Math. Pures Appl., IX. Ser. 36, 235-249 (1957; Zbl 0084.30402)]. It concerns principal \(G\)-bundles \(P\to W\) over a smooth oriented Riemannian 4-manifold with compact structure group. It says that anti-self dual connections on \(P\) which are gauge equivalent to infinite order at some point \(p\in W\) are locally gauge equivalent. This theorem was stated as part of Theorem 1.5 in the author's monograph [\(L^2\)-moduli spaces on 4-manifolds with cylindrical ends, Monographs in Geometry and Topology 1, International Press (Cambridge 1993; Zbl 0830.58004)] and the proof here corrects a gap in the preprint version of this monograph. The second theorem generalizes a result of \textit{S. K. Donaldson} for holomorphic bundles over Kähler manifolds [J. Geom. Phys. 8, No. 1-4, 89-122 (1992; Zbl 0747.53022)]. Here, \(W\) is a smooth, connected, Riemannian 4-manifold with compact, non-empty boundary \(\partial W\), \(M\) is a component of \(\partial W\), \(i: M\to \partial M\) is the inclusion, and \(P\to W\) is a principal bundle as before. Let \(A\), \(A'\) be anti-self dual connections on \(P\) which are Sobolev class \(L^2_3\) on some neighborhood of the boundary. The second theorem shows that if \(i^* A\) is gauge equivalent to \(i^* A'\) on \(M\), then \(A\) is locally gauge equivalent to \(A'\) on \(W\). Neither theorem follows from theorems about elliptic equations since the anti-self dual equations do not linearize as elliptic equations for the connection. The third theorem is an infinitesimal version of the first, and the last two theorems generalize a lemma in \textit{S. K. Donaldson} and \textit{P. B. Kronheimer} [The geometry of four-manifolds, Oxford: Clarendon Press (1990; Zbl 0820.57002)] and concern anti-self dual connections which are reducible on some subset.
gauge equivalent anti-selfdual connections
scientific article; zbMATH DE number 820206