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 o...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.展开更多
基金Supported by the National Natural Science Foundations of China(Grant Nos.1126103811361041)the Natural Science Foundation of Jiangxi Province(Grant No.20132BAB201005)
文摘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.
