  The purpose of this paper is to study the shapes and stabilities of
bio-membranes within the framework of exterior differential forms. After a
brief review of the current status in theoretical and experimental studies on
the shapes of bio-membranes, a geometric scheme is proposed to discuss the
shape equation of closed lipid bilayers, the shape equation and boundary
conditions of open lipid bilayers and two-component membranes, the shape
equation and in-plane strain equations of cell membranes with cross-linking
structures, and the stabilities of closed lipid bilayers and cell membranes.
The key point of this scheme is to deal with the variational problems on the
surfaces embedded in three-dimensional Euclidean space by using exterior
differential forms.
