  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.
