Lojasiewicz inequalities near simple bubble trees

In this paper we prove a gap phenomenon for critical points of the H-functional on
\nclosed non-spherical surfaces when H is constant, and in this setting furthermore prove
\nthat sequences of almost critical points satisfy Lojasiewicz inequalities as they approach
\nthe first non-trivial bubble tree.
\n
\nTo prove these results we derive sufficient conditions for Lojasiewicz inequalities to hold
\nnear a finite-dimensional submanifold of almost-critical points for suitable functionals on
\na Hilbert space.