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.
Citation
Yawei Chu. Xue Wang. "On the scalar curvature estimates for gradient Yamabe solitons." Kodai Math. J. 36 (2) 246 - 257, June 2013. https://doi.org/10.2996/kmj/1372337516
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