Abstract

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.

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