a b s t r a c t 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 m is finite. In this paper, we prove that (M n , g) is a constant curvature space if the L n 2 -norm of • R m 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 results 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.

We prove a best possible L p -estimate for compact CMC surfaces immersed into a 3-dimensional unit sphere S 3 , and as a consequence we obtain an L p -characterization of the flat torus in S 3 .