  The purpose of this paper is to give a self-contained exposition of the
Atiyah-Bott picture for the Yang-Mills equation over Riemann surfaces with an
emphasis on the analogy to finite dimensional geometric invariant theory. The
main motivation is to provide a careful study of the semistable and unstable
orbits: This includes the analogue of the Ness uniqueness theorem for
Yang-Mills connections, the Kempf-Ness theorem, the Hilbert-Mumford criterion
and a new proof of the moment-weight inequality following an approach outlined
by Donaldson. A central ingredient in our discussion is the Yang-Mills flow for
which we assume longtime existence and convergence.
