  Motivated by the study of the equilibrium equations for a soap film hanging
from a wire frame, we prove a compactness theorem for surfaces with
asymptotically vanishing mean curvature and fixed or converging boundaries. In
particular, we obtain sufficient geometric conditions for the minimal surfaces
spanned by a given boundary to represent all the possible limits of sequences
of almost-minimal surfaces. Finally, we provide some sharp quantitative
estimates on the distance of an almost-minimal surface from its limit minimal
surface.
