La géométrie de Bakry-Émery et l’écart fondamental
Séminaire de théorie spectrale et géométrie, Volume 28  (2009-2010), p. 147-157

This is a brief survey of recent results culminating in the proof of the fundamental gap conjecture by Andrews and Clutterbuck [1]. Recalling the Bakry-Émery geometry and Laplacian, we present our joint results with Z. Lu [14] 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.

Cet article est une présentation rapide, d’une part de résultats de l’auteur et Z. Lu [14], et d’autre part, de la résolution de la conjecture de l’écart fondamental par Andrews et Clutterbuck [1]. Nous commençons par rappeler ce qu’est la géométrie de Bakry-Émery, nous poursuivons en montrant les liens entre valeurs propres du laplacien de Dirichlet et de Neumann. Nous démontrons ensuite un rapport entre l’écart fondamental et la géométrie de Bakry-Émery, puis nous présentons les idées principales de la preuve de la conjecture de l’écart fondamental de [1]. Nous concluons par des résultats pour l’écart des triangles et des simplexes.

DOI : https://doi.org/10.5802/tsg.282
Classification:  35P05,  58J50
Keywords: fundamental gap, spectral gap, Dirichlet eigenvalues, Neumann eigenvalues, Bakry-Émery geometry, drift Laplacian, simplices
@article{TSG_2009-2010__28__147_0,
     author = {Rowlett, Julie},
     title = {La g\'eom\'etrie de Bakry-\'Emery et l'\'ecart fondamental},
     journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {28},
     year = {2009-2010},
     pages = {147-157},
     doi = {10.5802/tsg.282},
     language = {fr},
     url = {http://www.numdam.org/item/TSG_2009-2010__28__147_0}
}
Rowlett, Julie. La géométrie de Bakry-Émery et l’écart fondamental. Séminaire de théorie spectrale et géométrie, Volume 28 (2009-2010) , pp. 147-157. doi : 10.5802/tsg.282. http://www.numdam.org/item/TSG_2009-2010__28__147_0/

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