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Geometric symmetry groups, conservation laws and group-invariant solutions of the Willmore equation. (English) Zbl 1073.53008
Mladenov, Ivaïlo 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 (ISBN 954-84952-8-7/pbk). 246-265 (2004).
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. For the entire collection see [Zbl 1048.53002].
53A05Surfaces in Euclidean space
53C21Methods of Riemannian geometry, including PDE methods; curvature restrictions (global)
58E12Applications of variational methods to minimal surfaces
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