Abstract
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 m̈ the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that R m̈ goes to zero uniformly at infinity if for \(p\geq \frac n2\), the L p-norm of R m̈ 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 \(p\geq \frac n2\), R is positive and the L p-norm of R m̈ is pinched in [0, C 1), where C 1 is an explicit positive constant depending only on n, p, R and the Yamabe constant. 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 extends Theorem 1 of Hebey and M. Vaugon (J. Geom. Anal. 6, 531–553, 1996). This result is sharp, and we can precisely characterize the case of equality. In particular, when n = 4, we recover results by Gursky (Indiana Univ. Math. J. 43, 747–774, 1994; Ann. Math. 148, 315–337, 1998).
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The authors thank the referee for his helpful suggestions.
Supported by National Natural Science Foundations of China (11261038, 11361041), Jiangxi Province Natural Science Foundation of China (20171BAB201001).
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Fu, HP., Xiao, LQ. Some L p Rigidity Results for Complete Manifolds with Harmonic Curvature. Potential Anal 48, 239–255 (2018). https://doi.org/10.1007/s11118-017-9636-8
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DOI: https://doi.org/10.1007/s11118-017-9636-8