Hopf vesicles in S³(1)
by
Josu Arroyo
Universidad del País Vasco, Spain
Coauthors: Oscar J. Garay (Universidad del País Vasco)

Poster

A simple geometric model, proposed around 1740 by D. Bernoulli, sets the elastic curves as critical points of the total tension

ó õ \gamma  \kappa2     (1)

\kappa being the curvature of the curve \gamma. L. Euler classified in 1744 the plane elastic curves and in 1859 Kirchoff considered the problem under the Hamiltonian viewpoint. Recently some authors have studied the existence and classification of (closed) elastic curves in real space forms (Bryant and Griffiths, Langer and Singer, Koiso, ...). In this context, the critical points of

ó õ \gamma  (\kappa+\lambda)2     (2)

would correspond with the elastic curves whose position at rest is not ''straight'' but circular instead. We call them circular \lambda-elastic curves.

On the other hand, as it is well known since the times of Lagrange and Young, the mean curvature of a surface measures the tension it receives from the surrounding space. It was around 1810 that Sophie Germain in her study of the elastic plates, set the elastic energy E(S), of a surface S equal to the integral with respect to the surface area of an even, symmetric function of the principal curvatures of S. If we assume that the surface may be immersed in a 3-space form; that the energy is quadratic; but that the two sides of the elastic surface are distinguished (as in polymer interface) then the assumption on evenness may not be satisfied, and hence the elastic energy is of the form

E(S)= ó õ S  (a+b(H-co)2+cG)·dA     (3)

where dA is the area element and H and G are the mean and Gaussian curvature of the surface, respectively. Physically this formula is the Hooke's law, where b and c are the bending rigidities and a is the stretching of the surface or surface tension. An equilibrium elastic surface S constitutes a critical point for E(S), subject to the constraints of the problem. For instance, if we assume constraint on constant enclosed volume, we have the functional

M(S)= ó õ S  (b(H-co)2+cG)·dA+a ó õ S  dA+d ó õ \Omega  dV   (4)

which, when the ambient space is R³, is of fundamental importance in understanding the role of bending elasticity for both equilibrium shapes and shape fluctuations of fluid layers. Following a model of Helfrich, the equilibrium shape of a membrane is determined by the minimization of the shape (or curvature) energy which can be given by (4). Helfrich has discussed the bending elasticity of fluid membranes as formed by lipids and proposed that the shapes of vesicles and perhaps, red blood cells represent minima of the curvature energy. In this context, c_o is the spontaneous curvature which describes the effect of an asymmetry of the membrane or its environment and can be used to treat shape transitions of red blood cells by chemical agents. The first term of (4), \int_S(b(H-c_o)²+cG)·dA, is the elastic energy of the membrane or vesicle (vesicle=closed membranes). The second and third terms take account of the constraints on constant area and volume or they can represent actual work.

In order to obtain the mechanical equilibrium of the vesicle membrane we have to compute the critical points of the shape energy M(S), and therefore the Euler-Lagrange equations of the problem.

If b=0 (we disregard the bending energy) the energy is just a multiple of the area and then Euler-Lagrange equation of (4) reduces to the condition of surfaces of constant mean curvature. If additionally no constraint on the volume is assumed, then we obtain the minimal surfaces. If b > 0, the evenness assumption is taken ( that is c_o=0) and no constraint on the fixed volume is considered ( d=0 ), we can suppose by scaling that b=1 and also assume that a is equal to the curvature of the ambient 3-manifold, so a=-1, 0, 1. Then the energy is commonly denoted by W in this case and it is called the Willmore functional. If we are in the Euclidean space, then this choice means that there is no surface tension on S.

We want to study the existence of vesicles in S³(1).

For simplicity we consider the functional

ó õ S  (H+\lambda)2dv     (5)

and look for Hopf tori in S³(1) which are critical points of (5). First we can relate the Hopf tori which are minima for (5), that is, those which are a type of vesicles in S³(1) of spontaneous curvature \lambda, with the 2\lambda-"circular" elastic curves in S²(1/2):

A Hopf torus M_\gamma is a critical point of \int_M\gamma(\alpha-\lambda)²in S³(1) , if and only if, \gamma is a critical point of  \int_\gamma(\kappa-2\lambda)² in S²(1/2).

Observe that for \lambda = 0, one gets Willmore tori in S³(1) as liftings of closed free elastica in S²(1/2). This is the way in which Pinkall obtained the first examples of Willmore tori in R³ which were not stereographic projections of minimal surfaces in S³(1)). Thus we are led to the problem of finding closed \lambda-elastic curves in S²(1). We solve this by:

• First, we compute the Euler-Lagrange equations associated to the problem and solve them by using the Jacobi elliptic functions.
• with the aid of some ideas of Langer-Singer we set then a ''computable'' closedness condition.
• The above condition turns out to be difficult to check. Attending to the two types of solutions to the Euler-Lagrange equations we found, we are able to show the existence of closed solutions for the first type and numerically established the existence of closed solution of the second type.

Date received: June 2, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafh-07.