2016/06/21 by Lingzhong Zeng, Zeng, Lingzhong
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #math.AP #math.DG
paper · pdf · doi:10.48550/arxiv.1606.06429
60pages. arXiv admin note: text overlap with arXiv:1606.02589
arxiv created 2016/06/21 · openalex publication_date 2016/06/21 · arxiv updated 2016/06/22 · openalex created_date 2016/07/22 · openalex updated_date 2026/07/28
In this paper, we investigate eigenvalues of the Dirichlet problem and the closed eigenvalue problem of drifting Laplacian on the complete metric measure spaces and establish the corresponding general formulas. By using those general formulas, we give some upper bounds of consecutive gap of the eigenvalues of the eigenvalue problems, which is sharp in the sense of the order of the eigenvalues. As some interesting applications, we study the eigenvalue of drifting Laplacian on Ricci solitons, self-shrinkers and product Riemannian manifolds. We give the explicit upper bounds of the gap of the consecutive eigenvalues of the drifting Laplacian. Since eigenvalues is invariant in the sense of isometry, by the classifications of Ricci solitons and self-shrinkers, we give the explicit upper bounds for the consecutive eigenvalues of the drifting Laplacian on a large class metric measure spaces. In addition, we also consider the case of product Riemannian manifolds with certain curvature conditions and some upper bounds are obtained. Basing on the case of Laplace operator, we also present a conjecture as follows: all of the eigenvalues of the Dirichlet problem of drifting Laplacian on metric measure spaces satisfy: λk+1-λk≤(λ2-λ1)k(1)/(n).We note the conjecture is true in some special cases.