Complete Manifolds with Harmonic Curvature and Finite $L^p$-Norm Curvature

DOI：10.3770/j.issn:2095-2651.2017.03.011

 作者 单位 付海平 南昌大学数学系, 江西 南昌 330031 但萍萍 南昌大学数学系, 江西 南昌 330031 宋书林 江南大学物联网工程学院, 江苏 无锡 214122

令$(M^n, g)(n\geq3)$是具有调和曲率和正Yamabe 常数的$n$维完备黎曼流形. $R$和$\mathring{Rm}$ 分别为$M$的数量曲率和无迹黎曼曲率张量.本文的主要结论是对于$p\geq n$,若$\mathring{Rm}$的$L^{p}$范数是有限的,则$\mathring{Rm}$在无穷远处会一致的趋于零. 作为应用,证明了:如果$(M^n, g)$的$R$是正的且其$\mathring{Rm}$的$L^{p}$范数是有限的,那么该流形是紧致的;如果$(M^n, g)$是具有非负数量曲率的完备非紧流形且其$\mathring{Rm}$的$L^{p}$范数是有限的,那么该流形是数量平坦的.另外还得到了:对于$p\geq \frac n2$,如果$(M^n, g)$的$R$ 是正的且其$\mathring{Rm}$ 的$L^{p}$范数足够小的,那么该流形等距于球空间型.特别是,对于$p\geq n$,若$(M^n, g)$的$R$是正的且其$\mathring{Rm}$的$L^{p}$范数在$[0,C)$ 中,这里的$C$ 是一个仅与$n$, $p$, $R$和Yamabe常数有关的明确的常数,则该流形等距于球空间型.

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.