  We analyze the linear stability of the edge of a thin liquid metal layer
subject to a transverse high-frequency AC magnetic field. The layer is treated
as a perfectly conducting liquid sheet that allows us to solve the problem
analytically for both a semi-infinite geometry with a straight edge and a thin
disk of finite radius. It is shown that the long-wave perturbations of a
straight edge are monotonically unstable when the wave number exceeds some
critical value $k_c,$ which is determined by the surface tension and the linear
density of the electromagnetic force acting on the edge. The higher the density
of electromagnetic force, the shorter the critical wavelength. The
perturbations with wavelength shorter than the critical are stabilized by the
surface tension, whereas the growth rate of long wave perturbations reduces as
$\sim k$ for wave numbers $k\to 0$. Thus, there is the fastest growing
perturbation with the wave number $k_\max=2/3 k_c$. By applying the general
approach developed for the semi-infinite sheet, we find that a circular disk
becomes linearly unstable with respect to exponentially growing perturbation
with the azimuthal wave number $m=2$ when the magnetic Bond number exceeds
$Bm_c=3\pi$. The instability characteristics agree well with the experimental
data.
