Spectral geometry
Speaker: Anusha Bhattacharya, IISER Mohali
Host: Saikat Mazumdar
Title: Anusha Bhattacharya, IISER Mohali
Time, day and date: 3:00:00 PM – 4:00:00 PM, Friday, September 18
Venue: Online (https://us06web.zoom.us/j/84551518386?pwd=7Kv5mWUygLqQPCDE7dpoBq3vMutUbB.1)
Meeting ID: 845 5151 8386
Passcode: 607090
Abstract: A central problem in spectral geometry is to determine how the eigenvalues of the Laplacian on a Riemannian manifold depend on geometric quantities, such as curvature, diameter, volume, and injectivity radius. While uniform spectral estimates are well understood under sectional curvature bounds, considerably less is known when only Ricci curvature bounds are imposed. We address this question by deriving uniform estimates and spectral approximation results for the eigenvalues of the Laplacian using discretizations of the Riemannian manifold.
We consider the class of closed Riemannian n-manifolds with Ricci curvature and injectivity radius bounded below and diameter bounded above. In this class, we prove uniform estimates for the eigenvalues of the Hodge Laplacian and connection Laplacian acting on differential forms. This extends earlier work of Dodziuk and Lott, which required sectional curvature bounds. As an application, we derive a uniform upper bound on the spectral gap for non-compact manifolds with Ricci curvature bounded below. Under the additional assumption of an upper Ricci curvature bound, we consider weighted proximity graphs constructed from sufficiently dense points on the manifold. We then establish uniform convergence of the spectrum of the associated normalized graph Laplacians to those of the Laplace-Beltrami operator.
We further study compact Riemannian manifolds with smooth cylindrical boundary satisfying the same geometric bounds. On these manifolds, we construct graph Laplacians from sufficiently dense point samples on appropriate truncations and establish uniform convergence of the spectrum of the truncated graph Laplacian to the Dirichlet spectrum of the Laplace-Beltrami operator.