Hyperbolic Geometry and the Helfrich Functional

The Helfrich model is a fundamental tool for determining the morphology of biological membranes. We relate the geometry of an important class of its equilibria to the geometry of sessile and pendant drops in the hyperbolic space ${\mathbf{H}}^3$. When the membrane surface meets the ideal boundary of hyperbolic space, a modification of the regularized area functional is related to the construction of closed equilibria for the Helfrich functional in ${\mathbf{R}}^3$.