On -spectrum and τ-quasi-Einstein metric

In this paper, we study L f p -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 L f 2 -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 .