On the scalar curvature estimates for gradient Yamabe solitons (Q363256)

The authors derive scalar curvature estimates for gradient Yamabe solitons. Let \((M^n,g)\) be an \(n\)-dimensional complete noncompact Riemannian manifold. This manifold is called a gradient Yamabe soliton if there exists a smooth function \(f\) on \(M\) and \(\lambda \in {\mathbb R}\) such that \(Rg + \text{Hess}(f) = \lambda g\), where \(R\) is the scalar curvature of \((M^n,g)\). Define \(f_1\) by \(f = 2(n-1)f_1\) and assume that the Bakry-Emery Ricci tensor \(\text{Ric}_{f_1} = \text{Ric} + \text{Hess}(f_1)\) satisfies \(\text{Ric}_{f_1} \geq K\) for some \(K \in {\mathbb R}\). Let \(R_\star\) be the infimum of \(R\) over \(M\). The main result of the authors states:NEWLINENEWLINE(1) If \(M^n\) is shrinking (\(\lambda > 0\)), then \(0 \leq R_\star \leq \lambda\) and \(R > 0\) unless \(R \equiv 0\) and \(M^n\) is isometric to the Gauss soliton;NEWLINENEWLINE(2) If \(M^n\) is steady (\(\lambda = 0\)), then \(R_\star = 0\) and \(R > 0\) unless \(R \equiv 0\) and \(M^n\) is either trivial or isometric to a Riemannian product manifold;NEWLINENEWLINE(3) If \(M^n\) is expanding (\(\lambda < 0\)), then \(\lambda \leq R_\star \leq 0\) and \(R > \lambda\) unless \(R \equiv \lambda\) and \(M^n\) is either trivial or isometric to a warped product manifold.NEWLINENEWLINESome applications are given. The authors also obtain some scalar curvature estimates for quasi gradient Yamabe solitons.