  We investigate the convergence of phase fields for the Willmore problem away
from the support of a limiting measure $\mu$. For this purpose, we introduce a
suitable notion of essentially uniform convergence. This mode of convergence is
a natural generalisation of uniform convergence that precisely describes the
convergence of phase fields in three dimensions.
  More in detail, we show that, in three space dimensions, points close to
which the phase fields stay bounded away from a pure phase lie either in the
support of the limiting mass measure $\mu$ or contribute a positive amount to
the limiting Willmore energy. Thus there can only be finitely many such points.
  As an application, we investigate the Hausdorff limit of level sets of
sequences of phase fields with bounded energy. We also obtain results on
boundedness and $L^p$-convergence of phase fields and convergence from outside
the interval between the wells of a double-well potential. For minimisers of
suitable energy functionals, we deduce uniform convergence of the phase fields
from essentially uniform convergence.
