This paper starts from a nice application of the coupling method to a traditional topic: the estimation of spectral gap (the first non-trivial eigenvalue). Some new variational formulas for the lower bound of the spectral gap of Laplacian on manifold or elliptic operators in Rd or Markov chains are reported [10]; [15]; [16]. The new formulas are especially powerful for the lower bounds, they have no common points with the classical variational formula (which goes back to Lord S. J. W. Rayleigh (1877) or E. Fischer (1905) and is particularly useful for the upper bounds). No analog of the new formulas ever appeared before. The formulas enable us to recover or improve the main known results. This will be illustrated by a comparison of the new results with the known ones in geometry. Next, we will explain the mathematical tool for proving the results. That is, the trilogy of the recent development of the coupling theory: The Markovian coupling, the optimal Markovian coupling and the construction of distances for coupling. Finally, some related results and some problems for the further study are also mentioned. It is hoped that the paper could be readable not only for probabilists but also for geometers and analysts.

This is the first one of a series of three papers. They are partially surveys on three aspects: 1) explaining the main ideas of our recent application of the coupling method to the estimation of spectral gap, 2) introducing some more recent progress on the study on some related topics, 3) collecting some open problems for the further study. The technical details are often avoided in order to keep the paper to be readable at the graduate level.

This is the second one of a series of three papers. The ideas introduced in the last paper are used to study the estimate of spectral gap and four classes of typical eigenvalue problems on manifolds. The comparison with the known optimal estimates are given, some new progress is reported and some open problems are proposed.

This is the last one of a series of three papers. Here, we discuss six topics related to the spectral gap: the gradient estimate, the heat kernel and Harnack inequality, the logarithmic Sobolev inequality, the convergence in total variation, the algebraic convergence and the in finiteimensional case. The perturbation of spectral gap and the logarithmic Sobolev constant under a linear transform is given (Theorem 5). A new proof for computing the logarithmic Sobolev constant in a basic case is also presented (Theorem 7).

This note is devoted to study the exponential convergence rate in the total variation for reversible Markov processes by comparing it with the spectral gap. It is proved that in a quite general setup, with a suitable restriction on the initial distributions, the rate is bounded from below by the spectral gap. Furthermore, in the compact case or for birth-death processes or half-line diffusions, the rate is shown to be equal to the spectral gap.