On the scalar curvature estimates for gradient Yamabe solitons

Abstract

Let (Mn,g) be a gradient Yamabe soliton Rg + Hess f = λg with Ricf1 ≥ K (see (1.3) for f1) and λ, K $in$ R are constants. In this paper, it is showed that for gradient shrinking Yamabe solitons, the scalar curvature R > 0 unless R ≡ 0 and (Mn,g) is the Gaussian soliton, and for gradient steady and expanding Yamabe solitons, R > λ unless R ≡ λ and (Mn,g) is either trivial or a Riemannian product manifold. Replacing the assumptions Ricf1 ≥ K by R ≥ λ, we also derive the corresponding scalar curvature estimates. In particular, we show that any shrinking gradient Yamabe soliton with R ≥ λ must have constant scalar curvature R ≡ λ. Moreover, the lower bounds of scalar curvature for quasi gradient Yamabe solitons are obtained.