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.
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.
Mot clés : écart fondamental, valeurs propres du laplacian, valeurs propres Dirichlets, valeurs propres Neumann, géométrie Bakry-Émery, laplacien dérive, simplexes
Keywords: fundamental gap, spectral gap, Dirichlet eigenvalues, Neumann eigenvalues, Bakry-Émery geometry, drift Laplacian, simplices
@article{TSG_2009-2010__28__147_0,
author = {Julie Rowlett},
title = {La g\'eom\'etrie de {Bakry-\'Emery} et l{\textquoteright}\'ecart fondamental},
journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
pages = {147--157},
publisher = {Institut Fourier},
address = {Grenoble},
volume = {28},
year = {2009-2010},
doi = {10.5802/tsg.282},
language = {fr},
url = {https://proceedings.centre-mersenne.org/articles/10.5802/tsg.282/}
}
TY - JOUR AU - Julie Rowlett TI - La géométrie de Bakry-Émery et l’écart fondamental JO - Séminaire de théorie spectrale et géométrie PY - 2009-2010 SP - 147 EP - 157 VL - 28 PB - Institut Fourier PP - Grenoble UR - https://proceedings.centre-mersenne.org/articles/10.5802/tsg.282/ DO - 10.5802/tsg.282 LA - fr ID - TSG_2009-2010__28__147_0 ER -
%0 Journal Article %A Julie Rowlett %T La géométrie de Bakry-Émery et l’écart fondamental %J Séminaire de théorie spectrale et géométrie %D 2009-2010 %P 147-157 %V 28 %I Institut Fourier %C Grenoble %U https://proceedings.centre-mersenne.org/articles/10.5802/tsg.282/ %R 10.5802/tsg.282 %G fr %F TSG_2009-2010__28__147_0
Julie Rowlett. La géométrie de Bakry-Émery et l’écart fondamental. Séminaire de théorie spectrale et géométrie, Tome 28 (2009-2010), pp. 147-157. doi : 10.5802/tsg.282. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.282/
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