  The instability and periodic deformation of bilayer membranes during freezing
processes are studied as a function of the difference of the shape energy
between the high and the low temperature membrane states. It is shown that
there exists a threshold stability condition, bellow which a planar
configuration will be deformed. Among the deformed shapes, the periodic curved
square textures are shown being one kind of the solutions of the associated
shape equation. In consistency with recent expe rimental observations, the
optimal ratio of period and amplitude for such a texture is found to be
approximately equal to (2)^{1/2}\pi.
