摘要
Let （M^n, g） be an n-dimensional complete noncompact Riemannian manifold with harmonic curvature and positive Sobolev constant. In this paper, by employing an elliptic estimation method, we show that （M^n, g） is a space form if it has sufficiently small Ln/2-norms of trace-free curvature tensor and nonnegative scalar curvature. Moreover, we get a gap theorem for （M^n, g） with positive scalar curvature.
Let （M^n, g） be an n-dimensional complete noncompact Riemannian manifold with harmonic curvature and positive Sobolev constant. In this paper, by employing an elliptic estimation method, we show that （M^n, g） is a space form if it has sufficiently small Ln/2-norms of trace-free curvature tensor and nonnegative scalar curvature. Moreover, we get a gap theorem for （M^n, g） with positive scalar curvature.
基金
Acknowledgements The author would like to express his sincere thanks to the referees for valuable remarks. This work was supported in part by the National Natural Science Foundation of China （Grant No. 11071225）, the Natural Science Foundation of Anhui Provincial Education Department （Grant No. KJ2012A228）, and Project-2010JPKC07.
