Some L p Rigidity Results for Complete Manifolds with Harmonic Curvature

Abstract

Let (M n , g)(n ≥ 3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R andRm the scalar curvature and the trace-free Riemannian curvature tensor of M , respectively. The main result of this paper states thatRm goes to zero uniformly at infinity if for p ≥ n 2 , the L p -norm ofRm 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 ofRm is pinched in [0, C 1 ), where C 1 is an explicit positive constant depending only on n, p, R and the Yamabe constant.In particular, we prove an L p ( n 2 ≤ p < n−2 2 (1+ 1 − 4 n ))-norm ofRic pinching theorem for complete, simply connected, locally conformally flat Riemannian n(n ≥ 6)-manifolds with constant negative scalar curvature.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 improves Thereom 1.1 and Corollary 1 of E. Hebey and M. Vaugon [18]. This rsult is sharped, and we can precisely characterize the case of equality.

Four-manifolds with positive Yamabe constant

“…By Theorems 1.13 and 1.15 in [21], (M 4 ,g) is isometric to the round S 4 , the real projective space RP 4 , a manifold which is isometrically covered by S 1 × S 3 with the product metric, or a manifold which is isometrically covered by S 1 × S 3 with a rotationally symmetric Derdziński metric.…”

On Compact Riemannian Manifolds with Harmonic Weyl Curvature

“…Remark 1.8. This result had been proved by Gursky [16], Xiao and the first author [13]. Corollary 1.9.…”

“…Some isolation theorems of Weyl curvature tensor of positive Einstein manifolds are given in [7,14,17,20,29], when its L p -norm is small. Some scholars classify conformally flat manifolds satisfying some curvature L p -pinching conditions [6,12,13,17,27,32]. Recently, Tran [31] obtain two rigidity results for a closed Riemannian manifold with harmonic Weyl curvature, which are a generalization of Tachibana's theorem for non-negative curvature operator [30] and integral gap result which extends Theorem 1.10 for manifolds with harmonic curvature in [11].…”

Rigidity theorems for compact Bach-flat manifolds with positive constant scalar curvature

“…Recently, two rigidity theorems of the Weyl curvature tensor of positive Ricci Einstein manifolds are given in [4,11,12], which improve results due to [14,16,20]. The second author and Xiao have studied compact manifolds with harmonic curvature to obtain some rigidity results in [8,9,10]. Here when a Riemannian manifold satisfies δRm = {∇ l R i jkl } = 0, we call it a manifold with harmonic curvature.…”

“…Remark 1.7. In [10], Xiao and the second author proved that an n-dimensional compact locally conformally flat manifold (M n , g)(n ≥ 4) with positive constant scalar curvature is isometric to a quotient of the round S n , if M n |Ric|…”