Nonexistence of nontrivial quasi-Einstein metrics (Q447788)

The paper under review studies nonexistence of quasi-Einstein metrics using the gradient estimate method and the weak maximum principle. Motivated by Theorem A in [\textit{J. S. Case}, Pac. J. Math. 248, No. 2, 277--284 (2010; Zbl 1204.53032)], it is proved that if \((M^n, g)\) is an \(n\)-dimensional complete Riemannian manifold with \(\mathrm{Ric}_f \geq \lambda\) where \(\lambda \leq 0\) and \(\Delta_f f = \phi(f)\) for some \(\phi : \mathbb R \to \mathbb R\) satisfying NEWLINE\[NEWLINE\phi^\prime(t) + \frac{2}{n} \phi(t) + \lambda \geq 0,NEWLINE\]NEWLINE then \(M^n\) is Einstein and \(f\) is constant.NEWLINENEWLINEThe author also considers the complete gradient shrinking Ricci solitons and establishes that for an \(n\)-dimensional complete gradient shrinking Ricci soliton \((M^n, g, f)\), the condition \(f \in L^{\infty}(M^n, \mathrm{e}^{-f}, \mathrm{d}vol)\) is equivalent to \(|\nabla f| \in L^\infty(M^n, \mathrm{e}^{-f}, \mathrm{d}vol).\) Furthermore, if the normalized function \(\tilde f \leq 0\) and \(M^n\) has no boundary, then \(M^n\) is Einstein and \(f\) is constant where \(\Delta_f f = -2 \lambda \tilde f.\)