It is a fundamental task to estimate the eigenvalues of the drifting Laplacian in
geometric analysis. Here, we prove a general formula for the Dirichlet eigenvalue
problem of the drifting Laplacian. Using this formula, we establish some eigenvalue
inequalities of the drifting Laplacian on some important smooth metric measure
spaces, including Riemannian manifolds isometrically immersed in a Euclidean space,
gradient Ricci solitons, self-shrinkers, manifolds admitting certain special functions,
metric measure spaces with Bakry–Émery curvature conditions, and round
cylinders. We use new and interesting techniques to construct trial functions and
deal with some inequalities. For example, on gradient Ricci solitons, we
construct the test functions via Busemann functions associated with the
geodesics to give an intrinsic inequality and use geometric rigidity to determine
eigenvalues under looser conditions. Recall that singularity classification of Ricci
flows has experienced considerable development in geometric analysis, and
geometric analysts obtained many important and interesting results. We apply
some of them to estimate the eigenvalues of the drifting Laplacian in our
settings.
