| In this paper,we study eigenvalues for the bi-drifting Laplacian on the bounded domain in the complete noncompact Riemannian manifolds.If the Riemannian manifold is an n-dimensional hyperbolic space and the differential operator is an incomplete bi-drifting Laplacian,by establish ing a theorem of Barta type,we prove a universal inequality,which can be viewed as a rigidity result associated with variable.As an application,we investigate the asymptotic behavior of the eigenvalues with any order when the bounded domain tends to the hyperbolic space.Also,we obtain some eigenvalue inequalities for the radial bi-drifting Laplacian on the noncompact Rie-mannian manifold with pinching condition of sectional curvature.In particular,when the radial symmetric potential is a concave function satisfying certain linear growth condition with respect to the distance,we obtain an eigenvalue inequality,which is universal.Also,by controlling the bound for the distance function,we successfully establish an eigenvalue inequality of bi-drifting Laplacian without the radial symmetric assumption for the potential function and condition of the Bakry-Emery curvature.At the end of this paper also obtain estimates for the eigenvalues of the bi-drifting Laplacian on cigar soliton. |
