Eigenvalues of collapsing domains and drift Laplacians

By introducing a weight function to the Laplace operator, Bakry and \'Emery defined the "drift Laplacian" to study diffusion processes. Our first main result is that, given a Bakry-\'Emery 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 $\R^n$ and Neumann eigenvalues of domains in $\R^{n+1}$ 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-\'Emery manifolds}.