Gradient Flow Dynamics for Cell Membranes in the Canham–Helfrich Model

Abstract The most energetically efficient way that a deformed red blood cell regains equilibrium is mathematically described by the gradient flow of the Canham–Helfrich functional, including a spontaneous curvature and the conservation of surface area and enclosed volume. Using a recently discovered multiplicity inequality, we prove the global existence and convergence of smooth solutions for spheres and axisymmetric tori, provided that the initial energy lies below explicit thresholds.