Rigidity of noncompact complete Bach-flat manifolds

Abstract

Let ( M , g ) be a noncompact complete Bach-flat manifold with positive Yamabe constant. We prove that ( M , g ) is flat if ( M , g ) has zero scalar curvature and sufficiently small L 2 bound of curvature tensor. When ( M , g ) has nonconstant scalar curvature, we prove that ( M , g ) is conformal to the flat space if ( M , g ) has sufficiently small L 2 bound of curvature tensor and L 4 / 3 bound of scalar curvature.