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

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.

DOI : https://doi.org/10.5802/tsg.282
Classification : 35P05,  58J50
Mots clés : écart fondamental, valeurs propres du laplacian, valeurs propres Dirichlets, valeurs propres Neumann, géométrie Bakry-Émery, laplacien dérive, simplexes
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Rowlett, Julie. 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. http://www.numdam.org/articles/10.5802/tsg.282/

[1] Andrews, B.; Clutterbuck, J. Proof of the fundamental gap conjecture arXiv 1006.1686, (2010)

[2] Antunes, P.; Freitas, P. A numerical study of the spectral gap, J. Phys. A, Volume 41 (2008) no. 5, pp. 055201, 19 | MR 2433425 | Zbl 1142.35054

[3] Bakry, D.; Émery, M. Diffusions hypercontractives, Séminaire de probabilités, XIX, 1983/84 (Lecture Notes in Math.), Volume 1123, Springer, Berlin, 1985, pp. 177-206 | EuDML 113511 | Numdam | MR 889476 | Zbl 0561.60080

[4] van den Berg, M. On condensation in the free-boson gas and the spectrum of the Laplacian, J. Statist. Phys., Volume 31 (1983) no. 3, pp. 623-637

[5] Betcke, T.; Lu, Z.; Rowlett, J. The fundamental gap of triangles (en préparation) | MR 450480 | Zbl 0334.26009

[6] Brascamp, H. J.; Lieb, E. H. On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis, Volume 22 (1976) no. 4, pp. 366-389 | MR 768584 | Zbl 0551.53001

[7] Chavel, I. Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, 115, Academic Press Inc., Orlando, FL, 1984 (Including a chapter by Burton Randol, With an appendix by Jozef Dodziuk) | MR 65391 | Zbl 0051.28802

[8] Courant, R.; Hilbert, D. Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953 | MR 916759 | Zbl 0661.35062

[9] Kirsch, W.; Simon, B. Comparison theorems for the gap of Schrödinger operators, J. Funct. Anal., Volume 75 (1987) no. 2, pp. 396-410 | MR 1185270 | Zbl 0805.34080

[10] Lavine, R. The eigenvalue gap for one-dimensional convex potentials, Proc. Amer. Math. Soc., Volume 121 (1994) no. 3, pp. 815-821 | MR 1170358

[11] Li, P.; Treibergs, A. Applications of eigenvalue techniques to geometry, Contemporary geometry (Univ. Ser. Math.), Plenum, New York, 1991, pp. 21-52 | MR 573435 | Zbl 0441.58014

[12] Li, P.; Yau, S. T. Estimates of eigenvalues of a compact Riemannian manifold, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) (Proc. Sympos. Pure Math., XXXVI), Amer. Math. Soc., Providence, R.I., 1980, pp. 205-239 | MR 2016700 | Zbl 1038.53041

[13] Lott, J. Some geometric properties of the Bakry-Émery-Ricci tensor, Comment. Math. Helv., Volume 78 (2003) no. 4, pp. 865-883

[14] Lu, Z.; Rowlett, J. The fundamental gap preprint (2009), arXiv 1003.0191v1

[15] Lu, Z.; Rowlett, J. The fundamental gap conjecture on polygonal domains arXiv :0810.4937, (2008) | MR 2485469 | Zbl 1162.35059

[16] Ma, L.; Liu, B. Convex eigenfunction of a drifting Laplacian operator and the fundamental gap, Pacific J. Math., Volume 240 (2009) no. 2, pp. 343-361 | Numdam | MR 829055 | Zbl 0603.35070

[17] Singer, I. M.; Wong, B.; Yau, S.-T.; Yau, S. S.-T. An estimate of the gap of the first two eigenvalues in the Schrödinger operator, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Volume 12 (1985) no. 2, pp. 319-333 | MR 2237206 | Zbl 1105.53035

[18] Sturm, K.-T. On the geometry of metric measure spaces. I, Acta Math., Volume 196 (2006) no. 1, pp. 65-131 | MR 711491

[19] Villani, C. Topics in optimal transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003 | MR 1964483 | Zbl 1106.90001

[20] Wei, G.; Wylie, W. Comparison geometry for the Bakry-Emery Ricci tensor, J. Differential Geom., Volume 83 (2009) no. 2, pp. 377-405 | MR 2577473 | Zbl 1189.53036

[21] Yu, Q. H.; Zhong, J. Q. Lower bounds of the gap between the first and second eigenvalues of the Schrödinger operator, Trans. Amer. Math. Soc., Volume 294 (1986) no. 1, pp. 341-349 | MR 819952 | Zbl 0593.53030

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