题目: Spectrum theory on Riemannian manifolds
摘要: We will begin with the Weyl criterion characterizing the spectrum of Laplacians, and the new Weyl criterion I developed with Charalambous. As an application, we will compute the spectrum of the Laplacian on non-compact complete manifolds which do not have poles. In the second part of the lectures, we will study the relationship between Riemannian geometry, in particular the collapsing theory, with the spectrum theory. I will talk on the results of Fukaya and Cheeger-Colding on the continuity of the eigenvalues on collapsing family of compact Riemannian manifolds. Finally, we will talk about the continuity of spectrum of the Laplacian on differential forms.