Rigidity of complete noncompact bach-flat n-manifolds

Let (M-n, g) be a complete noncompact Bach-flat n-manifold with the positive Yamabe constant and constant scalar curvature. Assume that the L-2-norm of the trace-free Riemannian curvature tensor (R) over circlem is finite. In this paper, we prove that (M-n, g) is a constant curvature space if the L-n/2-norm of (R) over circlem is sufficiently small. Moreover, we get a gap theorem for (M-n, g) with positive scalar curvature. This can be viewed as a generalization of our earlier resuits of 4-dimensional Bach-flat manifolds with constant scalar curvature R >= 0 [Y.W. Chu, A rigidity theorem for complete noncompact Bach-flat manifolds, J. Geom. Phys. 61 (2011) 516-521]. Furthermore, when n > 9, we derive a rigidity result for R < 0. (C) 2012 Elsevier B.V. All rights reserved.