Автор: Hajime Urakawa
Название: Spectral Geometry of the Laplacian: Spectral Analysis and Differential Geometry of the Laplacian
Издательство: World Scientific Publishing Company
Год: 2017
ISBN: 9789813109087
Язык: English
Формат: pdf
Размер: 5,1 mb
Страниц: 312

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–P?lya–Weinberger's inequality of the Dirichlet eigenvalue of the Laplacian are also described. Then, the theorem of Colin de Verdi?re, 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.