Spectral Geometry of the Laplacian

The totality of the eigenvalues of the Laplacian of a compact Riemannian manifold is called 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 first eigenvalue, the Lichnerowicz–Obata's theorem on the first eigenvalue, the Cheng's estimates of the kth eigenvalues, and Payne–Pólya–Weinberger's inequality of the Dirichlet eigenvalue of the Laplacian are also described. Then, the theorem of Colin de Verdière, that is, the spectrum determines the totality of all the lengths of closed geodesics is described. We give the V Guillemin and D Kazhdan's theorem which determines the Riemannian manifold of negative curvature.

Contents:Fundamental Materials of Riemannian GeometryThe Space of Riemannian Metrics, and Continuity of the EigenvaluesCheeger and Yau Estimates on the Minimum Positive EigenvalueThe Estimations of the kth Eigenvalue and Lichnerowicz-Obata's TheoremThe Payne, Pólya and Weinberger Type Inequalities for the Dirichlet EigenvaluesThe Heat Equation and the Set of Lengths of Closed GeodesicsNegative Curvature Manifolds and the Spectral Rigidity Theorem
Readership: Researchers in differential geometry and partial differential equations.