Lipid bilayer vesicles minimize the Helfrich free energy F = ∫(κ/2)(2H)² dA + κ_G ∫K dA subject to fixed area and volume constraints. Explore how reduced volume ν and bending rigidity κ shape the membrane.
Sphere: ν=1 (F_bend minimal). Prolate ellipsoid: ν~0.65. Oblate: c₀ < 0. Stomatocyte: ν~0.3. Helfrich 1973 showed membranes behave like thin elastic shells with bending modulus κ~10–20 kT for DPPC.