Eigenvalue Comparison on Bakry-emery Manifolds

It is called shrinking, steady, or expanding soliton if a > 0, a = 0 or a < 0 respectively. More generally (M, g, f) is called a Bakry-Emery manifold if the so-called Bakry-Emery Ricci tensor Rcij + fij ≥ agij for some a ∈ R. In this paper we apply the modulus of continuity estimates developed in [AC,AC2,AC3] to give a different proof of an eigenvalue comparison estimate on Bakry-Emery manifolds for the operator ∆f + ∆ − ⟨∇(·),∇f⟩ on strictly convex Ω ⊂ M with diameter D and smooth boundary. This result was first proved in [BQ] (Theorem 14), which serves as a generalization to the earlier works of PayneWeinberger[PW], Li-Yau[LY] and Zhong-Yang [ZY]. An more recent result of this kind was obtained in [FS].