Microscopic Monte Carlo simulations of liquid sheets of copper and tin have been performed in order to study the dependence of the surface tension on the thickness of the sheet. It results that the surface tension is constant with the thickness as long as the sheet remains in one piece. When the sheet is getting thinner, holes start to appear, and the calculated surface tension rapidly decreases with thickness until the sheet becomes totally unstable and forms a cylinder. We assume here that this decrease is not due to a confinement effect as proposed by Werth et al. [Physica A392, 2359 (2013)] on Lennard-Jones systems, but to the appearance of holes that reduces the energy cost of the surface modification. We also show in this work that a link can be established between the stability of the sheet and the local fluctuations of the surface position, which directly depends on the value of the surface tension. Finally, we complete this study by investigating systems interacting through different forms of Lennard-Jones potentials to check if similar conclusions can be drawn.