Inverted catenoids, curvature singularities and tethered membranes

TL;DR

This paper explores the properties of inverted catenoids, revealing how curvature singularities and external forces influence the equilibrium shapes and stress distributions of tethered membranes, with implications for membrane shape transitions.

Contribution

It introduces the analysis of inverted minimal surfaces, highlighting the role of curvature singularities and external forces in membrane shape transitions.

Findings

Inverted catenoids exhibit curvature singularities at poles.

External forces pull poles together, affecting membrane stress.

Shape transitions occur with changing external forces and fixed surface area.

Abstract

If a catenoid is inverted in any interior point, a deflated compact geometry is obtained which touches at two points (its poles). The catenoid is a minimal surface and, as such, is an equilibrium shape of a symmetric fluid membrane. The conformal symmetry of the Hamiltonian implies that inverted minimal surfaces are also equilibrium shapes. However, they exhibit curvature singularities at their poles. These singularities are associated with external forces pulling the poles together. Unlike the catenoid which is free of stress, there will be stress within the inverted shapes. If the surface area is fixed, reducing the external force induces a transition from a discocyte to a cup-shaped stomatocyte.