  Here we prove the necessary analytic results to construct a Morse theory for
the Yang-Mills-Higgs functional on the space of Higgs bundles over a compact
Riemann surface. The main result is that the gradient flow with initial
conditions $(A'', \phi)$ converges to a critical point of this functional, the
isomorphism class of which is given by the graded object associated to the
Harder-Narasimhan-Seshadri filtration of $(A'', \phi)$. In particular, the
results of this paper show that the failure of hyperk\"ahler Kirwan
surjectivity for rank 2 fixed determinant Higgs bundles does not occur because
of a failure of the existence of a Morse theory.
