On -spectrum and τ-quasi-Einstein metric

In this paper, we study -spectrum estimates for some smooth potential function f on complete noncompact Riemannian manifolds via the τ-Bakry–Émery curvature. As its applications, we give the upper bound estimate of the -spectrum for the complete noncompact τ-quasi-Einstein metric, we give an example to show the sharpness of our estimate; we also get a lower bound estimate when λ<0, τ>1 and μ>0.