Urakawa H. Spectral Geometry of the Laplacian: Spectral Analysis and Differential Geometry of the Laplacian

Singapore: World Scientific Publishing, 2017. — 308 p.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–Polya–Weinberger's inequality of the Dirichlet eigenvalue of the Laplacian are also described. Then, the theorem of Colin de Verdiere, 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.Fundamental Materials of Riemannian Geometry
The Space of Riemannian Metrics, and Continuity of the Eigenvalues
Cheeger and Yau Estimates on the Minimum Positive Eigenvalue
The Estimations of the kth Eigenvalue and Lichnerowicz-Obata's Theorem
The Payne, Polya and Weinberger Type Inequalities for the Dirichlet Eigenvalues
The Heat Equation and the Set of Lengths of Closed Geodesics
Negative Curvature Manifolds and the Spectral Rigidity Theorem