Some Lp Rigidity Results for Complete Manifolds with Harmonic Curvature

• Published in: Potential analysis Vol. 48; no. 2; pp. 239 - 255
• Main Authors: Fu, Hai-Ping, Xiao, Li-Qun
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 2018
• Subjects: Functional Analysis; Geometry; Mathematics; Mathematics and Statistics; Potential Theory; Probability Theory and Stochastic Processes; 53C20; Harmonic curvature; 53C21; Constant curvature space; Trace-free curvature tensor
• ISSN: 0926-2601; 1572-929X
• DOI: 10.1007/s11118-017-9636-8

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 2 , the L p -norm of R m ̈ 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 ≥ n 2 , R is positive and the L p -norm of R m ̈ 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 ).