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

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