*(English)*Zbl 1073.53008

Willmore surfaces, namely the extremals of the functional $W\left(S\right)={\int}_{S}{H}^{2}\phantom{\rule{0.166667em}{0ex}}dA$, obey the Euler-Lagrange equation

$H$, $K$ respectively denoting the mean and the Gaussian curvatures of $S$.

The authors show that, in Mongé representation, equation (1), also called Willmore equation, can be regarded as a nonlinear fourth-order partial differential equation. This allows to prove that the symmetry group of the functional $W$ is the largest group of geometric transformations admitted by the Willmore equation.

The authors explicitely determine the conserved currents of ten linearly independent generators of the considered group. Then a special class of Willmore surfaces is studied, namely the rotationally-invariant solutions of (1).

The authors also point out some applications of the theory of Willmore surfaces in different areas, such as biophysics and 2D string theory.