General estimate of the first eigenvalue on manifolds

Abstract

Ten sharp lower estimates of the first non-trivial eigenvalue of Laplacian on compact Riemannian manifolds are reviewed and compared. An improved variational formula, a general common estimate, and a new sharp one are added. The best lower estimates are now updated. The new estimates provide a global picture of what one can expect by our approach.

Keywords

First non-trivial eigenvalue / sharp estimate / Riemannian manifold

References

[1] Bérard P. H., Besson G., Gallot S. Sur une inéqualité isopérimétrique qui généralise celle de Paul Lévy-Gromov. Invent Math, 1985, 80, 295-308

[2] Chen M. F. Optimal Markovian couplings and applications. Acta Math Sin (New Ser), 1994, 10 3 260-275

[3] Chen M. F. Explicit bounds of the first eigenvalue. Sci China (A), 2000, 43 10 1051-1059

[4] Chen M. F. Variational formulas and approximation theorems for the first eigenvalue. Sci China (A), 2001, 44 4 409-418

[5] Chen M. F. Eigenvalues, Inequalities, and Ergodic Theory, 2005, London: Springer.

[6] Chen M. F. Speed of stability for birth-death processes. Front Math China, 2010, 5 3 379-515

[7] Chen M. F., Scacciatelli E., Yao L. Linear approximation of the first eigenvalue on compact manifolds. Sci Sin (A), 2002, 45 4 450-461

[8] Chen M. F., Wang F. Y. Application of coupling method to the first eigenvalue on manifold. Sci Sin (A), 1993, 23 11 1130-1140

[9] Chen M. F., Wang F. Y. General formula for lower bound of the first eigenvalue on Riemannian manifolds. Sci Sin (A), 1997, 40 4 384-394

[10] Jia F. Estimate of the first eigenvalue of a compact Riemannian manifold with Ricci curvature bounded below by a negative constant. Chin Ann Math, 1991, 12A 4 496-502

[11] Lichnerowicz A. Géométrie des Groupes des Transformations. Paris: Dunod, 1958

[12] Ling J. Lower bounds of the eigenvalues of compact manifolds with positive Ricci curvature. Proc Amer Math Soc, 2006, 134 10 3071-3079

[13] Ling J. An exact solution to an equation and the first eigenvalue of a compact manifold. Illinois J Math, 2008, 51 3 853-860

[14] Schoen S., Yau S. T. Differential Geometry, 1988, Beijing: Science Press.

[15] Shi Y. M., Zhang H. C. Lower bounds for the first eigenvalue on compact manifolds. Chin Ann Math, 2007, 28A 6 863-866

[16] Wang F. Y. Functional Inequalities, Markov Processes, and Spectral Theory, 2004, Beijing: Science Press.

[17] Xu S. L., Pang H. D. Estimate of the first eigenvalue on compact manifolds. Math Appl, 2001, 14 1 116-119

[18] Xu S. L., Yang F. Y., Xu X. Estimate of the first eigenvalue of compact manifold with positive Ricci curvature. Math Appl, 2002, 15 2 85-88

[19] Yang H. C. Estimate of the first eigenvalue for a compact Riemannian manifold. Sci Sin (A), 1990, 33 1 39-51

[20] Zhao D. Eigenvalue estimate on a compact Riemannian manifold. Sci China (A), 1999, 42 9 897-904

[21] Zhong J. Q., Yang H. C. Estimates of the first eigenvalue of a compact Riemannian manifold. Sci Sin (A), 1984, 27 12 1265-1273