Mathematical Research Letters

Volume 19 (2012)

Number 3

Eigenvalues of collapsing domains and drift Laplacians

Pages: 627 – 648

DOI: http://dx.doi.org/10.4310/MRL.2012.v19.n3.a10

Authors

Zhiqin Lu (Department of Mathematics, University of California at Irvine)

Julie Rowlett (Max Planck Institut für Mathematik, Bonn, Germany)

Abstract

By introducing a weight function to the Laplace operator,Bakry and Émery defined the “drift Laplacian” to studydiffusion processes. Our first main result is that, givena Bakry–Émery manifold, there is a naturallyassociated family of graphs whose eigenvalues converge tothe eigenvalues of the drift Laplacian as the graphscollapse to the manifold. Applications of this resultinclude a new relationship between Dirichlet eigenvalues ofdomains in $\R^n$ and Neumann eigenvalues of domains in$\R^{n+1}$ and a new maximum principle. Using our mainresult and maximum principle, we are able to generalizeall the results in Riemannian geometry based ongradient estimates to Bakry–Émery manifolds.

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