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

Abstract

Let $(M^n,g)(n\ge 3)$ 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 $p\ge \frac{n}{2}$, 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 $p\ge \frac{n}{2}$, $R$ 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 $L^p(\frac{n}{2}\le p<\frac{n-2}{2}(1+\sqrt{1-\frac{4}{n}}))$-norm of $\mathring{Ric}$ pinching theorem for complete, simply connected, locally conformally flat Riemannian $n(n\ge 6)$-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.

We revise the older version, and add some contents

References in corpus (2)

• Rigidity Theorem for integral pinched shrinking Ricci solitons
• On conformally flat manifolds with constant positive scalar curvature