  In this work a non-trivial effect of the interfacial curvature on the
stability of accelerated interfaces, such as liquid rims, is uncovered. The new
stability analysis, based on operator and boundary perturbation theories,
reveals and quantifies influence of the interfacial curvature on the growth
rate and on the wavenumber selection of the Rayleigh-Taylor instability. The
systematic approach developed here also provides a rigorous generalization of
the widely used \textit{ad hoc} idea, due to Layzer [Astrophys. J.
\textbf{122}, 1-12 (1955)], of approximating the potential velocity field near
the interface.
