no. 6
A sharp lower bound for the first eigenvalue on Finsler manifolds
Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 6, pp. 983-996.

In this paper, we give a sharp lower bound for the first (nonzero) Neumann eigenvalue of Finsler-Laplacian in Finsler manifolds in terms of diameter, dimension, weighted Ricci curvature.

DOI : 10.1016/j.anihpc.2012.12.008
Classification : 35P15, 53C60, 35A23
Mots clés : Finsler-Laplacian, First eigenvalue, Weighted Ricci curvature, Poincaré inequality 
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     title = {A sharp lower bound for the first eigenvalue on {Finsler} manifolds},
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Wang, Guofang; Xia, Chao. A sharp lower bound for the first eigenvalue on Finsler manifolds. Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 6, pp. 983-996. doi : 10.1016/j.anihpc.2012.12.008. http://archive.numdam.org/articles/10.1016/j.anihpc.2012.12.008/

[1] D. Bao, S.S. Chern, Z. Shen, An Introduction to Riemann–Finsler Geometry, Springer-Verlag, New York (2000) | MR | Zbl

[2] R. Bartolo, E. Caponio, A.V. Germinario, M. Sánchez, Convex domains of Finsler and Riemannian manifolds, Calc. Var. Partial Differential Equations 40 no. 3–4 (2011), 335-356 | MR | Zbl

[3] D. Bakry, Z. Qian, Some new results on eigenvectors via dimension, diameter, and Ricci curvature, Adv. Math. 155 no. 1 (2000), 98-153 | MR | Zbl

[4] M. Belloni, V. Ferone, B. Kawohl, Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators, Z. Angew. Math. Phys. 54 (2003), 771-783 | MR | Zbl

[5] M. Chen, F. Wang, General formula for lower bound of the first eigenvalue on Riemannian manifolds, Sci. China Ser. A 40 no. 4 (1997), 384-394 | MR | Zbl

[6] Y. Ge, Z. Shen, Eigenvalues and eigenfunctions of metric measure manifolds, Proc. London Math. Soc. 82 (2001), 725-746 | MR | Zbl

[7] F. Hang, X. Wang, A remark on Zhong–Yangʼs eigenvalue estimate, Int. Math. Res. Not. no. 18 (2007) | MR

[8] P. Kröger, On the spectral gap for compact manifolds, J. Differential Geom. 36 no. 2 (1992), 315-330 | MR | Zbl

[9] P. Li, A lower bound for the first eigenvalue of the Laplacian on a compact manifold, Indiana Math. J. 28 (1979), 1013-1019 | MR | Zbl

[10] P. Li, S.-Y. Yau, 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. vol. XXXVI, Amer. Math. Soc., Providence, RI (1980)

[11] A. Lichnerowicz, Geometrie des groupes de transforamtions, Travaux et Recherches Mathemtiques vol. III, Dunod, Paris (1958)

[12] J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009), 903-991 | MR | Zbl

[13] M. Obata, Certain conditions for a Riemannian manifold to be a sphere, J. Math. Soc. Japan. 14 (1962), 333-340 | MR | Zbl

[14] S. Ohta, Finsler interpolation inequalities, Calc. Var. Partial Differential Equations 36 (2009), 211-249 | MR | Zbl

[15] S. Ohta, K.-T. Sturm, Heat flow on Finsler manifolds, Comm. Pure Appl. Math. 62 (2009), 1386-1433 | MR | Zbl

[16] S. Ohta, K.-T. Sturm, Bochner–Weitzenböck formula and Li–Yau estimates on Finsler manifolds, arXiv:1104.5276 | MR | Zbl

[17] L. Payne, H. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Ration. Mech. Anal. 5 (1960), 286-292 | MR | Zbl

[18] Z. Qian, H.-C. Zhang, X.-P. Zhu, Sharp Spectral Gap and Li–Yauʼs Estimate on Alexandrov Spaces, Math. Z., http://dx.doi.org/10.1007/s00209-012-1049-1.

[19] Z. Shen, Lectures on Finsler Geometry, World Scientific Publishing Co., Singapore (2001) | MR | Zbl

[20] K.-T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006), 65-131 | MR | Zbl

[21] G. Wang, C. Xia, An optimal anisotropic Poincaré inequality for convex domains, Pac. J. Math. 258 (2012), 305-326 | MR | Zbl

[22] B.Y. Wu, Y.L. Xin, Comparison theorems in Finsler geometry and their applications, Math. Ann. 337 (2007), 177-196 | MR | Zbl

[23] C. Villani, Optimal Transport, Old and New, Springer-Verlag (2009) | MR | Zbl

[24] J.Q. Zhong, H.C. Yang, On the estimate of the first eigenvalue of a compact Riemannian manifold, Sci. Sinica Ser. A 27 no. 12 (1984), 1265-1273 | MR | Zbl

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