A New Direction in Mathematics for Materials Science by Susumu Ikeda,Motoko Kotani

The totality of the eigenvalues of the Laplacian of a compact Riemannian manifold is named the spectrum. We describe how the spectrum determines a Riemannian manifold. The continuity of the eigenvalue of the Laplacian, Cheeger and Yau's estimate of the 1st eigenvalue, the Lichnerowicz–Obata's theorem at the first eigenvalue, the Cheng's estimates of the kth eigenvalues, and Payne–Pólya–Weinberger's inequality of the Dirichlet eigenvalue of the Laplacian also are defined.