Helfrich Functional in H2×R

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.