Hyperbolic Geometry and the Helfrich Functional

TL;DR

This paper explores the relationship between hyperbolic geometry and the Helfrich functional, providing new insights into membrane morphology and equilibrium shapes using hyperbolic space models.

Contribution

It establishes a novel connection between Helfrich membrane equilibria and hyperbolic geometric structures, extending the understanding of biological membrane shapes.

Findings

Relation between Helfrich equilibria and hyperbolic geometry

Modified area functional linked to membrane boundary conditions

Construction of closed equilibria in Euclidean space from hyperbolic models

Abstract

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 ${\bf H}^3$. 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 ${\bf R}^3$.