| 具有调和曲率与有限$L^p$曲率的完备流形 |
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| 引用本文： | 付海平,但萍萍,宋书林.具有调和曲率与有限$L^p$曲率的完备流形[J].数学研究及应用,2017,37(3):335-344. |
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| 作者姓名： | 付海平 但萍萍 宋书林 |
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| 作者单位： | 南昌大学数学系, 江西 南昌 330031,南昌大学数学系, 江西 南昌 330031,江南大学物联网工程学院, 江苏 无锡 214122 |
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| 基金项目： | 国家自然科学基金(Grant Nos.11261038; 11361041), 江西省自然科学基金(Grant No.20132BAB201005). |
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| 摘 要： | 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?m the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that R?m goes to zero uniformly at infinity if for p ≥ n, the L~p-norm of R?m is finite.As applications, we prove that(M~n, g) is compact if the L~p-norm of R?m 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 R?m. We prove that(M~n, g) is isometric to a spherical space form if for p ≥n/2, the L~p-norm of R?m 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 ≥ n, R is positive and the L~p-norm of R?m is pinched in 0, C), where C is an explicit positive constant depending only on n, p, R and the Yamabe constant.
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| 关 键 词： | 调和曲率 无迹曲率张量 常曲率空间 |
| 收稿时间： | 2016/1/25 0:00:00 |
| 修稿时间： | 2017/2/27 0:00:00 |
Complete Manifolds with Harmonic Curvature and Finite $L^p$-Norm Curvature |
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| Haiping FU,Pingping DAN and Shulin SONG.Complete Manifolds with Harmonic Curvature and Finite $L^p$-Norm Curvature[J].Journal of Mathematical Research with Applications,2017,37(3):335-344. |
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| Authors: | Haiping FU Pingping DAN and Shulin SONG |
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| Institution: | Department of Mathematics, Nanchang University, Jiangxi 330031, P. R. China,Department of Mathematics, Nanchang University, Jiangxi 330031, P. R. China and School of IOT Engineering, Jiangnan University, Jiangsu 214122, P. R. China |
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| 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. |
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| Keywords: | Harmonic curvature trace-free curvature tensor constant curvature space |
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