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.

10 pages, To appear J. Geom. Physics

Cited by in corpus (5)

• Fourth order curvature flows and geometric applications
• The tetralogy of Birkhoff theorems
• Universal curvature identities II
• Rigidity of complete Riemannian manifolds with vanishing Bach tensor
• Rigidity theorem for compact Bach-flat manifolds with positive constant ${\sigma }_2$