Functionals on Toroidal Surfaces
نویسندگان
چکیده
We show that the torus in R is a critical point of a sequence of functionalsFn (n = 1, 2, 3, . . .) defined over compact surfaces (closed membranes) in R. When the Lagrange function E is a polynomial of degree n of the mean curvatureH of the torus, the radii (a, r) of the torus are constrained to satisfy a 2 r2 = n−n n2−n−1 , n ≥ 2. A simple generalization of torus in R 3 is a tube of radius r along a curve α which we call it toroidal surface (TS). We show that toroidal surfaces with non-circular curve α do not provide minimal energy surfaces of the functionals Fn (n = 2, 3) on closed surfaces. We discuss possible applications of the functionals discussed in this work on cell membranes. MSC: 53C42, 53A10, 49Q05, 49Q10, 74K15
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