Estimates for Eigenvalues of the Dirichlet Laplacian on Riemannian Manifolds

Abstract

The authors 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, they derive upper and lower bounds for eigenvalues. For the projective spaces and their minimal submanifolds, they also give explicit estimates on the lower bound for the eigenvalue of the Dirichlet Laplacian.

Keywords

Laplacian / Eigenvalues / Weyl’s law / Riesz mean / Universal estimates / 53C42 / 58J50

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