Event № 246
We consider the drifting Laplacian over a noncompact, smooth, weighted manifold. We associate to the weighted manifold a family of higher dimensional Riemannian manifolds in warped product form. We show that various geometric analysis results on the weighted manifold are closely related to those on the warped product, by directly relating the geometry of the two spaces. In particular, we can demonstrate Gaussian heat kernel estimates for the drifting Laplacian over the weighted manifold whenever its Bakry-Emery Ricci tensor is bounded below. These are obtained effortlessly from the respective heat kernel bounds on the warped product. The proofs reveal the strong geometric connection of the weighted space to the warped product spaces. At the same time, they further illustrate the fact that the drifting Laplacian and Bakry-Emery Ricci tensor are projections (in some sense) of the Laplacian and Ricci tensor of a higher dimensional space. We then use these results to study the spectrum of the drifting Laplacian on the weighted manifold. This is joint work with Zhiqin Lu.