  This paper surveys the main results obtained during the period 1992-1999 on
three aspects mentioned at the title. The first result is a new and general
variational formula for the lower bound of spectral gap (i.e., the first
non-trivial eigenvalue) of elliptic operators in Euclidean space, Laplacian on
Riemannian manifolds or Markov chains (\S 1). Here, a probabilistic
method-coupling method is adopted. The new formula is a dual of the classical
variational formula. The last formula is actually equivalent to Poincar\'e
inequality. To which, there are closely related logarithmic Sobolev inequality,
Nash inequality, Liggett inequality and so on. These inequalities are treated
in a unified way by using Cheeger's method which comes from Riemannian
geometry. This consists of \S 2. The results on these two aspects are mainly
completed by the author joint with F. Y. Wang. Furthermore, a diagram of the
inequalities and the traditional three types of ergodicity is presented (\S 3).
The diagram extends the ergodic theory of Markov processes. The details of the
methods used in the paper will be explained in a subsequent paper under the
same title.
