  We consider the thermal undulation, or shape fluctuation, of an almost planar
fluid membrane surrounded by the same near-critical binary fluid mixtures on
both sides. A weak preferential attraction is assumed between the membrane and
one component of the mixture. We use the Gaussian free-energy functional to
study the equilibrium average of the undulation amplitude within the linear
approximation with respect to the amplitude. According to our result given by a
simple analytic formula, the ambient near-criticality tends to suppress the
undulation of a membrane, and this suppression effect can overwhelm that of the
bending rigidity for small wave numbers. Thus, the ambient near-criticality is
suggested to prevent a large membrane from becoming floppy even if the lateral
tension vanishes at the equilibrium.
