This paper presents a complete analysis of the Helfrich membrane energy functional in the product space H2×R. We address the analytical challenges posed by the ideal boundary of the space by developing a renormalization scheme, allowing us to formulate a well-posed variational problem. We derive the Euler-Lagrange equations for the renormalized functional, characterizing the equilibrium configurations through a coupled system of partial differential equations and a Neumann-type boundary condition. A central result of our work is a rigidity theorem, proven via a Killing field argument, which establishes that any admissible critical surface is necessarily axially symmetric. Finally, we connect this mathematical theory to biophysics by proposing a new variational principle for the Solvent Accessible Surface (SAS) under geometric confinement, demonstrating that our classified surfaces represent the optimal elastic energy shapes for such systems.
Nieto et al. (Mon,) studied this question.