Some L ^p Rigidity Results for Complete Manifolds with Harmonic Curvature

Abstract

Let (M ⁿ, g)(n ≥ 3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R and R m̈ the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that R m̈ goes to zero uniformly at infinity if for $p\geq \frac n2$, the L ^p-norm of R m̈ is finite. Moreover, If R is positive, then (M ⁿ, g) is compact. As applications, we prove that (M ⁿ, g) is isometric to a spherical space form if for $p\geq \frac n2$, R is positive and the L ^p-norm of R m̈ is pinched in [0, C ₁), where C ₁ is an explicit positive constant depending only on n, p, R and the Yamabe constant. We give an isolation theorem of the trace-free Ricci curvature tensor of compact locally conformally flat Riemannian n-manifolds with constant positive scalar curvature, which extends Theorem 1 of Hebey and M. Vaugon (J. Geom. Anal. 6, 531–553, 1996). This result is sharp, and we can precisely characterize the case of equality. In particular, when n = 4, we recover results by Gursky (Indiana Univ. Math. J. 43, 747–774, 1994; Ann. Math. 148, 315–337, 1998).