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Zbl 1073.53008
Vassilev, Vassil M.; Mladenov, Iva\"ilo M.
Geometric symmetry groups, conservation laws and group-invariant solutions of the Willmore equation.
(English)
[A] Mladenov, Iva\"ilo M. (ed.) et al., Proceedings of the 5th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 5--12, 2003. Sofia: Bulgarian Academy of Sciences. 246-265 (2004). ISBN 954-84952-8-7/pbk

Willmore surfaces, namely the extremals of the functional $W(S)=\int_S H^2 \,dA$, obey the Euler-Lagrange equation $$\Delta H+ 2(H^2- K)H= 0,\tag1$$ $H$, $K$ 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.
[Maria Falcitelli (Bari)]
MSC 2000:
*53A05 Surfaces in Euclidean space
53C21 Methods of Riemannian geometry (global)
58E12 Appl. of variational methods to minimal surfaces

Keywords: Willmore equation; Mongé representation

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