ESTIMATES FOR THE EIGENVALUES OF THE DRIFTING LAPLACIAN ON SOME COMPLETE RICCI SOLITONS

Abstract

Ricci solitons are the self-similar solutions to the Ricci flow, which play an important role in understanding the singularity dilations of the Ricci flow. In this paper, we investigate eigenvalues of the Dirichlet problem of a drifting Laplacian on some important complete Ricci solitons: the product shrinking Ricci soliton, cigar soliton, and so on. Since eigenvalues are invariant of isometries, we can give the estimates for the eigenvalues of a drifting Laplacian on the rotationally invariant shrinking solitons. In addition, we also obtain a sharp upper bound of the kth eigenvalue of the a drifting Laplacian on the product Ricci soliton in the sense of order k.