Upper bounds on the first eigenvalue for a diffusion operator via Bakry-Émery Ricci curvature
Let \((M,g)\) be an \(n\)-dimensional complete Riemannian manifold and let \(\phi\) be a \(C^2\) function. Consider the symmetric diffusion operator \(L=\Delta-\nabla\phi\cdot\phi\). This is the infinitesimal generator of the Dirichlet form associated to \(d\mu:=e^{-\phi}dx\). It is self-adjoint with respect to the weighted measure \(d\mu\). Let \(Ric_{m,n}\) be the associated Bakry-Emery Ricci curvature of the diffusion operator \(L\) where \(m\) is the Bakry-Emery dimension of \(L\) [\textit{D. Bakry} and \textit{M. Emery}, Proc., Lect. Notes Math. 1123, 177--206 (1985; Zbl 0561.60080)] -- this is not necessarily an integer. Assume that \(Ric_{m,n}\geq-(n-1)\). Then the authors establish an upper bound estimate on the first nontrivial eigenvalue of the diffusion operator \(L\) of the form \[ \lambda_1\leq(m-1)(n-1)/4\,. \] This generalizes a previous result of \textit{S.-Y. Cheng} [Math. Z. 143, 289--297 (1975; Zbl 0329.53035)] for the Laplacian to the case of the diffusion operator.
scientific article; zbMATH DE number 5625382
scientific article