A gap theorem for complete four-dimensional manifolds with $\deltaW^{+} = 0$

Abstract

Let $M^{4}$ be a complete noncompact oriented fourdimensional Riemannian manifold satisfying $\delta W^{+}=0$, where $W^{+}$ is the self-dual part of the Weyl curvature tensor. Suppose its scalar curvature is nonnegative and Sobolev's inequality holds. We show that if the $L^{2}$ norm of $W^{+}$ is sufficiently small, then $W^{+}\equiv 0$.