Morse theory for the space of Higgs bundles

The purpose of this paper is to 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 converges to a critical point of this functional, the isomorphism class of which is given by the graded object associated to theHarder–Narasimhan–Seshadri filtration of the initial condition. In particular,the results of this paper show that the failure of hyperkahler Kirwan surjectivity for rank 2 fixed determinant Higgs bundles does not occur because of a failure of the existence of a Morse theory.