Complete Manifolds with Harmonic Curvature and Finite $L^p$-Norm 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 $p\geq n$, the $L^{p}$-norm of $\mathring{Rm}$ is finite. As applications, we prove that $(M^n, g)$ is compact if the $L^{p}$-norm of $\mathring{Rm}$ is finite and $R$ is positive, and $(M^n, g)$ is scalar flat if $(M^n, g)$ is a complete noncompact manifold with nonnegative scalar curvature and finite $L^{p}$-norm of $\mathring{Rm}$. We prove that $(M^n, g)$ is isometric to a spherical space form if for $p\geq \frac n2$, the $L^{p}$-norm of $\mathring{Rm}$ is sufficiently small and $R$ is positive. In particular, we prove that $(M^n, g)$ is isometric to a spherical space form if for $p\geq n$, $R$ is positive and the $L^{p}$-norm of $\mathring{Rm}$ is pinched in $[0,C)$, where $C$ is an explicit positive constant depending only on $n, p$, $R$ and the Yamabe constant.