FUNCTIONALS ON TOROIDAL SURFACES

We show that the torus in R-3 is a critical point of a sequence of functionals F-n (n = 1, 2, 3,...) defined over compact surfaces (closed membranes) in R-3. When the Lagrange function epsilon 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, a(2)/r(2) = n(2) - n/n(2)-n-1, n >= 2. A simple generalization of torus in R-3 is a tube of radius r along a curve alpha which we call it toroidal surface (TS). We show that toroidal surfaces with non-circular curve alpha do not provide minimal energy surfaces of the functionals F-2 (n = 2,3) on closed surfaces. We discuss possible applications of the functionals discussed in this work on cell membranes.