On the structure of conformally flat Riemannian manifolds

$L^2$ curvature pinching theorems and vanishing theorems on complete Riemannian manifolds

“…Considering the case of p = 1 in Theorem 4.3, using an analogous method as above, we have Remark 6.1. In [14], H.Z. Lin proved a one-end theorem for LCF manifolds by assuming that R ≤ 0 and ( M |E| n dv) 2 n < C(n) for some explicit constant C(n) > 0.…”

“…In [22], Pigola, Rigoli and Setti showed a vanishing result for bounded harmonic forms of middle degree on complete non-compact LCF manifolds, by adding suitable conditions on scalar curvature and volume growth. In [14], Lin proved some vanishing and finiteness theorems for L 2 harmonic 1-forms on complete non-compact LCF manifolds under integral curvature pinching conditions.…”

$L^{p}$ harmonic 1-forms on conformally flat Riemannian manifolds

“…For L 2 harmonic forms, Lin [23] proved some vanishing and finiteness theorems for L 2 harmonic 1-forms on a locally conformally flat Riemannian manifold that satisfies an integral pinching condition on the traceless Ricci tensor and for which the scalar curvature is nonpositive or satisfies some integral pinching conditions. Dong et al [10] proved van-ishing theorems for L 2 harmonic p-forms on a complete noncompact locally conformally flat Riemannian manifold under suitable conditions.…”

“…[23]) Let (M m , g) be an m-dimensional complete Riemannian manifold. ThenRic ≥ -|T|g -|R| √ m gin the sense of quadratic forms, where Ric, R, and T = Ric -R m g are the Ricci curvature tensor, the scalar curvature, and the traceless Ricci tensor of (M m , g), respectively.…”

“…[23]) Let (M m , g) (m ≥ 3) be an m-dimensional complete, simply connected, and locally conformally flat Riemannian manifold with R ≤ 0 or M |R| m 2 dv < ∞. Then we have the…”