Estimates for Eigenvalues of the Dirichlet Laplacian on Riemannian Manifolds

We revisit the eigenvalue problem of the Dirichlet Laplacian on bounded domains in complete Riemannian manifolds. By building on classical results like Li-Yau's and Yang's inequalities, we derive upper and lower bounds for eigenvalues. For the projective spaces and their minimal submanifolds, we also give explicit estimates on lower bounds for eigenvalues of the Dirichlet Laplacian.