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Register a new userLa geometrie de Bakry-Emery et lecart fondamental
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Authors
Julie Rowlett
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This article is a brief presentation of results surrounding the fundamental gap. We begin by recalling Bakry-Emery geometry and demonstrate connections between eigenvalues of the Laplacian with the Dirichlet and Neumann boundary conditions. We then show a connection between the fundamental gap and Bakry-Emery geometry, concluding with a presentation of the key ideas in Andrewss and Clutterbucks proof of the fundamental gap conjecture. We conclude with a presentation of results for the fundamental gap of triangles and simplices.
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We demonstrate lower bounds for the eigenvalues of compact Bakry-Emery manifolds with and without boundary. The lower bounds for the first eigenvalue rely on a generalised maximum principle which allows gradient estimates in the Riemannian setting to be directly applied to the Bakry-Emery setting. Lower bounds for all eigenvalues are demonstrated using heat kernel estimates and a suitable Sobolev inequality.
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In this paper, we first prove the $f$-mean curvature comparison in a smooth metric measure space when the Bakry-Emery Ricci tensor is bounded from below and $|f|$ is bounded. Based on this, we define a Myers-type compactness theorem by generalizing the results of Cheeger, Gromov, and Taylor and of Wan for the Bakry-Emery Ricci tensor. Moreover, we improve a result from Soylu by using a weaker condition on a derivative $f(t)$.
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