Global existence and convergence of solutions to gradient systems and applications to Yang-Mills flow

We discuss our results on global existence and convergence of solutions to the gradient flow equation for the Yang-Mills energy functional over a closed, four-dimensional, Riemannian manifolds: If the initial connection is close enough to a minimum of the Yang-Mills energy functional, in a norm or energy sense, then the Yang-Mills gradient flow exists for all time and converges to a Yang-Mills connection. The Lojasiewicz-Simon gradient inequality plays an essential role in our approach and we describe its role in global existence and convergence problems and in proofs of discreteness of energies for Yang-Mills connections and harmonic maps.