Rigidity of complete non-compact generalized $m$-quasi-Einstein manifolds

Abstract

We study complete non-compact gradient generalized m-quasi-Einstein manifolds with constant scalar curvature $R \le 0$, soliton function $λ> 0$, and $m > 1$, where the coefficient $μ= 1/m$ is constant. We introduce the weighted function $v = e^{-f/m}λ$ and prove it is subharmonic. This leads to five rigidity results, each forcing the manifold to be Euclidean. We first show by a concrete example that if $μ$ is allowed to be nonconstant, the rigidity conclusions fail even when all other hypotheses are satisfied. Therefore the constant mu condition is essential.