References
Chen, M. F., Wang, F. Y., General formula for lower bound of the first eigenvalue on Riemannian manifolds.Sci. Sin., 1997, 40(4): 384.
Zhong, J. Q., Yang, H. C., Estimates of the first eigenvalue of a compact Riemannian manifolds,Sci. Sin., 1984, 27 (12): 1251.
Lichnerowicz, A.,Géométrie des Groupes des Transformations, Dunod, Paris, 1958.
Cai, K. R., Estimate on lower bound of the first eigenvalue of a compact Riemannian manifold,Chin. Ann. of Math., 1991, 12(B)(3): 267.
Yang, H. C., Estimate of the first eigenvalue of a compact Riemannian manifold with Ricci curvature bounded below by a negative constant,Sci. Sin. (in Chinese), 1989, 32(A)(7): 689.
Jia, F., Estimate of the first eigenvalue of a compact Riemannian manifold with Ricci curvature bounded below by a negative constantChin. Ann. Math. (in Chinese), 1991, 12(A): 496.
Bérard, P. H., Besson, G., Gallot, S., Sur une inéqualité isopérimétrique qui generalise celle de Paul Lévy-Gromov,Invent. Math., 1985, 80: 295.
Cheeger, J., A lower bound for the smallest eigenvalue of the Laplacian, inProblems in analysis, a symposium in honor of S. Bochner, Princeton: Princeton U. Press, 1970, 195–199.
Li, P., Yau, S. T., Estimates of eigenvalue of a compact Riemannian manifold,Ann. Math. Soc. Proc. Symp. Pure. Math., 1980, 36: 205.
Bakry, D., Ledoux, M., Levy-Gromov’s isoperimetric inequality for an infinite dimensional diffusion generator,Invent Math., 1996, 123: 259.
Bérard, P. H., Spectral geometry: direct and inverse problem,LNM., Springer-Verlag, Vol. 1207, 1986.
Bobkov, S., A functional form of the isoperimetric inequality for the Gaussian measure,J. Funct. Anal., 1996, 135(1): 39.
Chavel, I.,Eigenvalues in Riemannian Geometry, Beijing: Academic Press, 1984.
Ledoux, M., Isoperimetry and Gaussian Analysis,Ecole d’ été de Probabilités de Saint-Flour 1994, to appear.
Li, P.,Lecture Notes on Geometric Analysis, Seoul: National Univ., 1993.
Saloff-Coste, L., Lectures on Finite Markov Chains,Ecole d’ été de Probabilités de Saint-Flour 1996, to appear.
Yau, S. T., Schoen, R.,Differential Geometry (in Chinese), Beijing: Science Press, 1988.
Singer, M., Wong, B., Yau, S. S. T., An estimate of the gap of the first two eigenvalues in the Schrödinger operator,Ann. Scuola Norm. Sup. Pisa, Ser. IV, 1985, XI(2): 319.
Wang, F. Y., Application of coupling method to the Neumann eigenvalue problem,Prob. Th. Rel. Fields, 1994, 98: 299.
Berestycki, H., Nirenberg, L., Varadhan, S. R. S., The principal eigenvalue and maximum principle for second-order elliptic operators in general domains,Comm, Pure and Appl., 1994, XLVII: 47.
Escobar, J. F., Uniqueness theorems on conformal deformation of metrics, Sobolev inequalities, and an eigenvalue estimate,Comm. Pure and Appl. Math., 1990, XLIII: 857.
Chen, R., Neumann eigenvalue estimate on a compact Riemannian manifold,Proc. Amer. Math. Soc., 1990, 108(4): 961.
Wang, F. Y., Estimate of the first Dirichlet eigenvalue by using the diffusion processes,Prob. Th. Rel. Fields, 1994, 101: 363.
Wang, F. Y., A probabilistic approach to the first Dirichlet eigenvalue on noncompact Riemannian manifold,Acta Math. Sin., New Ser., 1997, 1197, 13(1): 116.
Wang, F. Y., Estimation of the first eigenvalue and the lattice Yang-Mills fields,Chin. J. Math., 1996, 17(2): 119.
Buser, P., Colbois, B., Dodziuk, J., Tubes and eigenvalues for negatively curved manifolds,J. Geom. Anal., 1993, 3(1): 1.
Tonno, S., The first eigenvalue of the Laplacian on spheres,Tôhoku Math. J., 1979, 31: 179.
Li, P., Tarn, L. F., Harmonic functions and the structure of complete manifolds,J. Diff. Geom., 1992, 35: 359.
Wang, X. H., Bounded harmonic functions on a class of complete Riemannian manifolds,Acta Math, Sin. (in Chinese), 1995, 38(2): 171.
Baum, H., Eigenvalue estimates for Dirac operators coupled to instantons,Annals of Global Anal. Geom., 1994, 12: 193.
Kirchberg, K. D., Compact six-dimensional Káhler spin manifolds of positive scalar curvature with the smallest possible first eigenvalue of the Dirac operator,Math. Ann., 1988, 282: 157.
Kirchberg, K. D., The first eigenvalue of the Dirac operator on Kähler manifolds,J. Geom. Phys., 1990, 7(4): 449.
Lu, K. P., The (0, 1) heat form of Bn and its application,Acta Math. Sin. (in Chinese), 1994, 37(2): 160.
Lu, K. P., The heat kernel of unitary group and its application,Acta Math. Sin (in Chinese), 1994, 37(6): 744.
Li, P., Tian, G., On the heat kernel of the Bergmann metric on algebraic varieties,J. Amer. Math. Soc., 1995, 8(4): 857.
Li, P., Schoen, R., Lp and mean value properties of subharmonic functions on Riemannian manifolds,Acta Math., 1984, 153: 279.
Lu, Y. G., An estimate on nonzero eigenvalues of Laplacian in nonlinear version, 1996.
Hoh, W., The martingale problem for a class of pseudo differential operators,Math. Ann., 1994, 300: 121.
Hoh, W., Pseudo differential operators with negative definite symbols and the martingale problem,Stochastics and Stoch. Reports, 1995, 55: 225.
Asada, S., Notes of eigenvalues of Laplacian acting on p-forms,Hokkaido Math. J., 1979, 8: 220.
Asada, S., On the first eigenvalue of the Laplacian acting on p-forms,Hokkaido Math. J., 1980, 9: 112.
Tonno, S., The spectrum of the Laplacian for 1-forms,Proc. Amer. Math. Soc, 1974, 45(1): 125.
Chung, F. R. K., Spectral graph theory,CBMS Lecture Notes, 1996.
Chung, F. R. K., Yau, S. T., A Harnack inequality for homogeneous graphs and subgraphs,Comm. Anal. Geom., 1994, 2: 628.
Chung, F. R. K., Yau, S. T., Logarithmic Harnack inequality,Math. Research Letters, 1997.
Chung, F. R. K., Graham, R. L., Yau, S. T., On sampling with Markov chains,Random Structures and Algorithm, 1996, 9: 55.
Chung, F. R. K., Graham, R. L., Yau, S. T., Eigenvalues and diameters for manifolds and graphs,Advances, 1997.
Diaconis, P., Saloff-Coste, L., Comparison theorems for reversible markov chains,Ann. Appl. Prob., 1993, 3(3): 696.
Diaconis, P., Saloff-Coste L., Nash inequality for finite Markov chains,J. Theor. Prob., 1996, 459–510.
Diaconis, P., Saloff-Coste, L., Logarithmic Sobolev inequality for finite Markov chains,Ann. Appl. Prob., 1996.
Diaconis, P., Stroock, D. W., Geometric bounds for eigenvalues of Markov chains,Ann. Appl. Prob., 1991, 1(1): 36.
Ingrassia, S., On the rate of convergence of the metropolis algorithm and Gibbs sampler by geometric bounds,Ann. Appl. Prob., 1994, 4(2): 347.
Jerrum, M. R., Sinclair, A. J., Approximating the permanent,SIAM J. Comput., 1989, 18: 1149.
Lawler, G. F., Sokal, A. D., Bounds on the L2 spectrum for Markov chain and Markov processes: a generalization of Cheeger’s inequality,Trans. Amer. Math. Soc., 1988, 309: 557.
Sullivan, W. G., The L2 spectral gap of certain positive recurrent Markov chains and jump processes,Z. Wahrs., 1994, 67: 387.
Thomas, L. E., Bound on the mass gap for finite volume stochastic Ising models at low temperature,Comm. Math. Phys., 1989, 126: 1.
Stong, R., Eigenvalues of random walks on groups,Ann. Prob., 1995, 23(4): 1961.
Sinclair, A. J., Jerrum, M. R., Approximate counting, uniform generation, and rapidly mixing Markov chains,Inform, and Comput., 1989, 82: 93.
Coulhon, T., Grigor’ yan, A., On diagonal lower bounds for heat kernels and Markov chains,Duke Univ. Math. J., 1997.
Chen, M. F., On ergodic region of Schlögl’s model, inDirichlet Forms and Stock. Proc. (eds. Ma, Z. M., Röckner, M., Yan, J. A.et al.), 1995, 87.
Author information
Authors and Affiliations
About this article
Cite this article
Chen, M. Coupling, spectral gap and related topics (II). Chin.Sci.Bull. 42, 1409–1416 (1997). https://doi.org/10.1007/BF02883046
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02883046