Universal inequalities of the poly-drifting Laplacian on the Gaussian and cylinder shrinking solitons

Abstract

In this paper, we study the eigenvalue problem of poly-drifting Laplacian and get a general inequality for lower order eigenvalues on compact smooth metric measure spaces with boundary (possibly empty). Applying this general inequality, we obtain some universal inequalities for lower order eigenvalues for the eigenvalue problem of poly-drifting Laplacian on bounded connected domains in Euclidean spaces or unit spheres. Moreover, we separately get some universal inequalities for the eigenvalue problem of poly-drifting Laplacian on bounded connected domains in the Gaussian and cylinder shrinking solitons.

Estimates for eigenvalues of weighted Laplacian and weighted $p$-Laplacian

“…Clearly, the first eigenvalue l 0 has multiplicity one and constant eigenfunction. In recent years, there are some interesting results concerning eigenvalue estimates of the drifting Laplacian and the bi-drifting Laplacian-see, e.g., [9,11,12,15,18,21,22,23,30]. When ðM; h ; iÞ is immersed into the Euclidean N-space ðR N ; canÞ with the canonical metric can, one can define the weighted mean curvature vector as ? , where H is the mean curvature vector of M in R N , ð'f Þ ?…”