Functionals on Closed 2-Surfaces

Abstract

We show that the 2-torus in ${\mathbb R}^3$ is a critical point of a sequence of functionals ${\cal F}_{n}$ ($n=1,2,3, \cdots$) defined over compact 2-surfaces in ${\mathbb R}^3$. When the Lagrange function ${\cal E}$ is a polynomial of degree $n$ of the mean curvature $H$ of the surface, the radii ($a,r$) of the 2-torus are related as $\frac{a^2}{r^2}=\frac{n^2-n}{n^2-n-1}, n \ge 2$. If the Lagrange function depends on both mean and Gaussian curvatures, the 2- torus remains to be a critical point of ${\cal F}_{n}$ without any constraints on the radii of the torus.