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The geometry of Bakry-Émery and the fundamental gap. (La géométrie de Bakry-Émery et l’écart fondamental.) (French. English summary) Zbl 1223.58027
Actes de Séminaire de Théorie Spectrale et Géométrie. Année 2009–2010. St. Martin d’Hères: Université de Grenoble I, Institut Fourier. Séminaire de Théorie Spectrale et Géométrie 28, 147-157 (2010).
Summary: This is a brief survey of recent results culminating in the proof of the fundamental gap conjecture by B. Andrews and J. Clutterbuck [J. Am. Math. Soc. 24, No. 3, 899–916 (2011; Zbl 1222.35130)]. Recalling the Bakry-Émery geometry and the Laplacian, we present our results with Z. Lu [“Eigenvalues of collapsing domains and drift Laplacians”, arXiv: 1003.0191], which demonstrate an intimate connection between the first non-trivial eigenvalue of a certain Bakry-Émery Laplacian and the fundamental gap. This is a special case of our more general results relating Dirichlet and Neumann eigenvalues and Bakry-Émery eigenvalues. Ideas particularly germane to the recent proof of the fundamental gap conjecture are discussed. In conclusion, we present recent results for the fundamental gap on the moduli spaces of $$n$$-simplices in general and triangles in particular.
For the entire collection see [Zbl 1213.35007].

##### MSC:
 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 35P05 General topics in linear spectral theory for PDEs
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