Functionals on Toroidal Surfaces

Abstract

We show that the torus in ${\mathbb R}^3$ is a critical point of a sequence of functionals ${\mathcal F}_{n}$ ($n=1,2,3, \ldots$) defined over compact surfaces (closed membranes) in ${\mathbb R}^3$. When the Lagrange function ${\mathcal E}$ is a polynomial of degree $n$ of the mean curvature $H$ of the torus, the radii ($a,r$) of the torus are constrained to satisfy $\frac{a^2}{r^2}=\frac{n^2-n}{n^2-n-1},~~ n \ge 2$. A simple generalization of torus in ${\mathbb R}^3$ is a tube of radius $r$ along a curve ${\bf \alpha}$ which we call it toroidal surface (TS). We show that toroidal surfaces with non-circular curve ${\bf \alpha}$ do not provide minimal energy surfaces of the functionals ${\mathcal F}_{n}$ ($n=2,3$) on closed surfaces. We discuss possible applications of the functionals discussed in this work on cell membranes.