A gap theorem for complete four-dimensional manifolds with \(\delta W^+=0\) (Q820184)

Let \(M^4\) be a complete, noncompact, oriented, Riemannian manifold of dimension \(4\). Denote by \(W^+\) the self-dual part of the Weyl curvature tensor. In this paper it is assumed that the formal divergence \(\delta W^+\) is null. Manifolds satisfying \(\delta W^+=0\) are natural generalizations of Einstein manifolds or symmetric spaces. The object of this paper is a gap theorem which shows that if, additionally, the scalar curvature of the manifold is nonnegative and Sobolev's inequality holds on \(M^4\), then there is a constant \(C>0\), depending only on the Sobolev constant, such that if \(\int_M | W^+| ^2 dV < C\), then \(W^+ \equiv 0\).