  In this work, we derive a sharp condition on the mass of the initial data for
the global existence of the Chern-Simons-Schr\"odinger equation. As a
corollary, we prove that if the strength of interaction is less than the
Bogomolny bound, then, for a large enough mass of initial data, there exists a
globally defined solution. On the other hand, for the interactions which are
above the Bogomolny bound, the critical mass condition on the initial data for
the global existence depends on the strength of the self-interacting field.
Then, we show that the states with the initial critical mass and zero energy
are standing wave solutions and globally well-posed. Moreover, they are static
if the self-interacting field is large enough and non-static for small
self-interacting field.
