  Two canonical formulations of the Einstein gravity in 2+1 dimensions, namely,
the ADM formalism and the Chern-Simons gravity, are investigated in the case of
nonvanishing cosmological constant. General arguments for reducing phase spaces
of the two formalisms are given when spatial hypersurface is compact. In
particular when the space has the topology of a sphere $S^{2}$ or a torus
$T^{2}$, the spacetimes constructed from these two formulations can be
identified and the classical equivalence between the ADM and the CSG is shown.
Moreover in the $g=1$ case the relations between their phase spaces, and
therefore between their quantizations, are given in almost the same form as
that in the case when the cosmological constant vanishes. There are, however,
some modifications, the most remarkable one of which is that the phase space of
the CSG is in 1 to 2 correspondence with the one of the ADM when the
cosmological constant is negative.
