  We prove Lojasiewicz inequalities for the harmonic map energy for maps from
surfaces of positive genus into general analytic target manifolds which are
close to simple bubble trees and as a consequence obtain new results on the
convergence of harmonic map flow and on the energy spectrum of harmonic maps
with small energy.
  Our results and techniques are not restricted to particular targets or to
integrable settings and we are able to lift general Lojasiewicz-Simon
inequalities valid near harmonic maps $\hat \omega:S^2\to N$ to the singular
setting whenever the bubble $\hat \omega$ is attached at a point which is not a
branch point.
