  We study the shapes of pored membranes within the framework of the Helfrich
theory under the constraints of fixed area and pore size. We show that the mean
curvature term leads to a budding- like structure, while the Gaussian curvature
term tends to flatten the membrane near the pore; this is corroborated by
simulation. We propose a scheme to deduce the ratio of the Gaussian rigidity to
the bending rigidity simply by observing the shape of the pored membrane. This
ratio is usually difficult to measure experimentally. In addition, we briefly
discuss the stability of a pore by relaxing the constraint of a fixed pore size
and adding the line tension. Finally, the flattening effect due to the Gaussian
curvature as found in studying pored membranes is extended to two-component
membranes. We find that sufficiently high contrast between the components'
Gaussian rigidities leads to budding which is distinct from that due to the
line tension.
