For (almost) critical points of geometric variational problems one often only has weak, rather than strong, compactness results. As a consequence there are many situations where the seminal results of Simon on Lojasiewicz inequalities are not applicable since sequences of almost critical points can form singularities or converge to a limit with a different topology. In this talk we consider this problem for almost harmonic maps from surfaces into analytic manifolds. We present a method, which is based on joint work with A. Malchiodi and B. Sharp, that allows us to prove Lojasiewicz inequalities for sequences that converge to a simple bubble tree, and as a result obtain new conclusions about the energy spectrum and the convergence of harmonic map flow for low energy maps from surfaces of positive genus. As we shall discuss, our method is not restricted to integrable settings, but allows us to lift general Lojasiewicz-Simon inequalities from the regular setting to the singular setting of simple bubble trees whenever the bubble is attached at a non-branched point.
