  We study the elastic properties of a two-dimensional fluctuating surface
whose area density is allowed to deviate from its optimal (Schulman) value. The
behavior of such a surface is determined by an interplay between the
area-dependent elastic energy, the curvature elasticity, and the entropy. We
identify three different elastic regimes depending on the ratio $A_p/A_s$
between the projected (frame) and the saturated areas. We show that thermal
fluctuations modify the elastic energy of stretched surfaces ($A_p/A_s> 1$),
and dominate the elastic energy of compressed surfaces ($A_p/A_s< 1$). When
$A_p\sim A_s$ the elastic energy is not much affected by the fluctuations; the
frame area at which the surface tension vanishes becomes smaller than $A_s$ and
the area elasticity modulus increases.
