Estimates for the eigenvalues of the bi-drifting Laplacian on complete metric measure spaces

Abstract

As we know, it is very difficult to prove an eigenvalue inequality of Ashbaugh–Cheng–Ichikawa–Mametsuka type for the bi-drifting Laplacian on the bounded domain of the complete metric measure spaces. Even assumed that the differential operator is bi-Laplacian on the n-dimensional unit Euclidean sphere, this problem still remains open. However, a general inequality with respect to the bi-drifting Laplacian is successfully established under certain conditions in this paper. Applying the general inequality, we prove some eigenvalue inequalities of Ashbaugh–Cheng–Ichikawa–Mametsuka type on the Gaussian shrinking Ricci soliton and the n-dimensional cigar metric measure spaces (CMMS for short). In particular, we obtain two interesting eigenvalue inequalities of Ashbaugh–Cheng–Ichikawa–Mametsuka type for the CMMS with lower-dimensional topology.