Łojasiewicz inequalities near simple bubble trees

abstract: In this paper we prove a gap phenomenon for critical points of the $H$-functional on closed non-spherical surfaces when $H$ is constant, and in this setting furthermore prove that sequences of almost critical points satisfy {\L}ojasiewicz inequalities as they approach the first non-trivial bubble tree. To prove these results we derive sufficient conditions for {\L}ojasiewicz inequalities to hold near a finite-dimensional submanifold of almost-critical points for suitable functionals on a Hilbert space.