Abstract
The present paper is concerned with the geometric Lie symmetry groups of the Willmore and shape equations—the Euler-Lagrange equations associated with the Willmore and Helfrich functionals. The ten-parameter group of special conformal transformations in the three-dimensional Euclidean space, which in known to be the symmetry group of the Willmore functional, is recognized as the largest group of geometric transformations admitted by these equations in Monge representation. The conserved currents of ten linearly independent conservation laws, which correspond to the variational symmetries of the Willmore equation and hold on its smooth solutions, are derived. The shape equation is found to admit only a six-parameter subgroup of the aforementioned ten-parameter group. Each symmetry admitted by the shape equation is its variational symmetry as well and the corresponding conserved currents are obtained.
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Vassilev, V.M., Djondjorov, P.A., Mladenov, I.M. (2014). Lie Group Analysis of the Willmore and Membrane Shape Equations. In: Ganghoffer, JF., Mladenov, I. (eds) Similarity and Symmetry Methods. Lecture Notes in Applied and Computational Mechanics, vol 73. Springer, Cham. https://doi.org/10.1007/978-3-319-08296-7_7
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