Eigenvalue Estimates for Beltrami-Laplacian Under Bakry-Émery Ricci Curvature Condition

On closed Riemannian manifolds with Bakry-Émery Ricci curvature bounded from below and gradient of the potential function, we obtain lower bounds for all positive eigenvalues Beltrami-Laplacian instead weighted Laplacian. The bound k th eigenvalue depends on k, curvature, dimension diameter upper manifold, but volume manifold is not involved.