Eigenvalues of collapsing domains and drift Laplacians

Abstract

By introducing a weight function to the Laplace operator, Bakry and Émery defined the "drift Laplacian" to study diffusion processes. Our first main result is that, given a Bakry-Émery manifold, there is a naturally associated family of graphs whose eigenvalues converge to the eigenvalues of the drift Laplacian as the graphs collapse to the manifold. Applications of this result include a new relationship between Dirichlet eigenvalues of domains in ${\mathbb{R}}^n$ and Neumann eigenvalues of domains in ${\mathbb{R}}^{n+1}$ and a new maximum principle. Using our main result and maximum principle, we are able to generalize \emph{all the results in Riemannian geometry based on gradient estimates to Bakry-Émery manifolds}.

References in corpus (2)

• Spectral simplicity and asymptotic separation of variables
• Lower diameter bounds for compact shrinking Ricci solitons

Cited by in corpus (5)

• Dirichlet eigenvalue sums on triangles are minimal for equilaterals
• On the first eigenvalue of the Witten-Laplacian and the diameter of compact shrinking Ricci solitons
• Lower diameter bounds for compact shrinking Ricci solitons
• Heat kernel estimates and the essential spectrum on weighted manifolds
• Some topics on Ricci solitons and self-similar solutions to mean curvature flow