Four-manifolds with positive Yamabe constant

Abstract

We refine Theorem B due to Gursky [26]. As applications, we give some rigidity theorems on four-manifolds with postive Yamabe constant. In particular, these rigidity theorems are sharp for our conditions have the additional properties of being sharp. By this we mean that we can precisely characterize the case of equality. We prove some classification theorems of four manifolds according to some conformal invariants (see Theorems 5.1, 1.3 and 1.6), which reprove and generalize the conformally invariant sphere theorem of Chang-Gursky-Yang [11], i.e., Theorem D.2000 Mathematics Subject Classification. Primary 53C21; Secondary 53C20.

On Compact Manifolds with Harmonic Curvature and Positive Scalar Curvature

“…Since g has constant scalar curvature, g is Einstein from the proof of Obata Theorem (see [23]). Hence 1 6 M R 2 = 1 6 µ([g]) 2 , by Corollary 1.16 in [13], we complete the proof of this corollary.…”

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

“…Combining (2.15) with (2.16), we get thatRic = 0, i.e., M n is an Einstin manifold. We finish the proof of Theorem By Theorem 1.8 in [8], we obtain that M 4 is isometric to a quotient of the round S 4 . We finish the proof of Theorem 1.1.…”

“…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.…”

On Compact Riemannian Manifolds with Harmonic Weyl Curvature

“…By Theorem 1.8 in [10] (see also [11]), we obtain that M 4 is isometric to a quotient of the round S 4 or a CP 2 with the Fubini-Study metric. Then we complete the proof of Corollary 1.4.…”