Nonexistence results for extremals of curvature functionals (Q1383710)

The author considers variational integrals of the form \[ F(X)= \int_X f(H, K)dA, \] where \(H\) and \(K\) denote mean and Gaussian curvature of the immersed surface \(X:\overline B\to \mathbb{R}^3\) defined on the closed unit disc \(\overline B\) in \(\mathbb{R}^2\). The integrand \(f\) is given by \(f(H, K)= \alpha+ \beta(H- H_0)^2- \gamma K\) with constants \(\alpha, \beta, \gamma\geq 0\) and \(H_0\in\mathbb{R}\). It is shown that any tubular surface solving the Euler equation for \(F\) is a circular cylinder, a torus or a surface consisting of pieces of cylinders and tori, and a smooth ruled surface \(S\) cannot satisfy the Euler equation for \(\beta H_0\neq 0\) unless \(S\) is locally cylindrical.