Equilibrium red blood shape configurations to Canham-Hilfrich energy: Analytical study

In this study, we revisit the Canham-Helfrich energy to analytically determine the equilibrium configurations of the red blood cell (RBC). By extending the work of Au and Wan (2003), we establish a sufficient analytical condition ensuring the existence of the biconcave equilibrium shape. This result provides a rigorous complement to previous asymptotic arguments, clarifying the mechanical balance underlying RBC morphology. Moreover, the analytical derivation of the equilibrium shape function from the Helfrich energy naturally reproduces the empirical contour equation proposed by Evans and Fung (1972). This demonstrates that their experimental results can be obtained via the energy-minimization framework, thereby linking experimental observations with a consistent theoretical foundation. The present results provide both a mathematical completion and a physical unification of previous models, confirming that the biconcave RBC geometry arises as an equilibrium configuration governed by simple curvature constraints.