  The mechanical effects of membrane compositional inhomogeneities are analyzed
in a process analogous of neck formation in cellular membranes. We cast on the
Canham-Helfrich model of fluid membranes with both the spontaneous curvature
and the surface tension being non-homogeneous functions along the cell
membrane. The inhomogeneous distribution is determined by the equilibrium
mechanical equations, and, in order to establish the role played by the
inhomogeneity, we focus on the catenoid, a surface of zero mean curvature,
which can be described in terms of the catenary curve parameterized by arc
length. We show that analytic solutions exist for the spontaneous curvature, as
well as for both, the surface tension and the radial elastic force. An analytic
expression for the constrictive force at the neck, is obtained. From the
energetic analysis, it is found that, if we fix the value of the constrictive
force at the neck, the set of solutions lies on two branches separated by an
energetic barrier. This barrier corresponds to the energy of the maximum
catenoid. If instead we fix the axial force, the solution has access to
catenoid of any size
