Some $L^p$ rigidity results for complete manifolds with harmonic curvature

Abstract

Let $(M^n, g)(n\geq3)$ be an $n$-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by $R$ and $\mathring{Rm}$ the scalar curvature and the trace-free Riemannian curvature tensor of $M$, respectively. The main result of this paper states that $\mathring{Rm}$ goes to zero uniformly at infinity if for $p\geq \frac n2$, the $L^{p}$-norm of $\mathring{Rm}$ 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 $\mathring{Rm}$ is pinched in $[0,C_1)$, where $C_1$ is an explicit positive constant depending only on $n, p$, $R$ and the Yamabe constant. In particular, we prove an $L^{p}(\frac n2\leq p<\frac{n-2}{2}(1+\sqrt{1-\frac4n}))$-norm of $\mathring{Ric}$ pinching theorem for complete, simply connected, locally conformally flat Riemannian $n(n\geq 6)$-manifolds with constant negative scalar curvature. 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 improves Thereom 1.1 and Corollary 1 of E. Hebey and M. Vaugon \cite{{HV}}. This rsult is sharped, and we can precisely characterize the case of equality.