Nonexistence of nontrivial quasi-Einstein metrics

Abstract

Let (Mⁿ, g, e^–f dvol_g) be a smooth metric measure space of dimension n. In this note, we first prove a nonexistence result for Mⁿ with the Bakry-Émery Ricci tensor is bounded from below. Then we show that f $\in$ L^∞ (Mⁿ, e^–f dvol) and |∇f| $\in$ L^∞ (Mⁿ, e^–f dvol) are equivalent for complete gradient shrinking Ricci solitons. Furthermore, we prove that there is no non-Einstein shrinking soliton when the normalized function $\tilde f$ is non-positive.