# American Institute of Mathematical Sciences

January  2014, 13(1): 453-481. doi: 10.3934/cpaa.2014.13.453

## Lie algebra symmetry analysis of the Helfrich and Willmore surface shape equations

 1 Department of Mathematics and Information Sciences, Northumbria University, Pandon Building, Camden Street, Newcastle upon Tyne, NE2 1XE, United Kingdom 2 INdAM-COFUND Marie Curie Fellow, Mathematisches Institut, Friedrich-Schiller-Universität, Jena, 07737, Germany

Received  April 2013 Revised  April 2013 Published  July 2013

We compute the Lie symmetry algebra of the equation of Helfrich surfaces and we show that it is the algebra of conformal vector fields of $R^2$. We also show that in the particular case of the Willmore surfaces we have to add the homothety vector field of $R^3$ to the aforementioned algebra. We prove that a Helfrich surface that is invariant w.r.t. a conformal symmetry is a helicoid and that all such surface solutions satisfy one and the same system of ordinary differential equations obtained by symmetry reduction. We also show that for the Willmore surface shape equation the symmetry reduction leads to two systems of ODEs. Then we construct explicit solutions in the case of revolution surfaces. The results obtained can be extended to the study of PDE problems in $2$ spatial dimensions admitting conformal Lie symmetries.
Citation: Giovanni De Matteis, Gianni Manno. Lie algebra symmetry analysis of the Helfrich and Willmore surface shape equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 453-481. doi: 10.3934/cpaa.2014.13.453
##### References:
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Han, Conditions for a Diffeomorphism to be embedded in a $C^r$ flow,, Acta Math. Sinica (N.S.), 4 (1988), 111. doi: 10.1007/BF02560593. Google Scholar [9] W. Helfrich, Elastic properties of lipid bilayers: Theory and possible experiments,, Z. Naturforsch, 28c (1973), 693. Google Scholar [10] B. G. Konopelchenko, On solutions of the shape equation for membranes and strings,, Phys. Lett. B, 414 (1997), 58. doi: 10.1016/S0370-2693(97)01137-4. Google Scholar [11] H. Jian-Guo and O.-Y. Zhong-can, Shape equations of the axisymmetric vesicles,, Phys. Rev. E, 47 (1993), 461. doi: 10.1103/PhysRevE.47.461. Google Scholar [12] P. F. Lam, Embedding homeomorphisms in differential flows,, Colloq. Math., 35 (1976), 275. Google Scholar [13] P. F. Lam, Embedding a differentiable homeomorphism in a flow subject to a regularity condition on the derivatives of the positive transition homeomorphisms,, J. Differential Equations, 30 (1978), 31. doi: 10.1016/0022-0396(78)90021-9. Google Scholar [14] P. F. Lam, Embedding homeomorphisms in $C^1$-flows,, Ann. Mat. Pura Appl., 123 (1980), 11. doi: 10.1007/BF01796537. Google Scholar [15] R. A. Leo, L. Martina and G. Soliani, Group analysis of the three-wave resonant system in $(2+1)$-dimensions,, J. Math. Phys., 27 (1986), 2623. doi: 10.1063/1.527280. Google Scholar [16] R. Lipowsky and E. Sackman, "Structure and Dynamics of Membranes,", Elsevier Science B.V., (1995). Google Scholar [17] G. Manno and R. Vitolo, Geometric aspects of higher order variational principles on submanifolds,, Acta Appl. Math., 101 (2008), 215. doi: 10.1007/s10440-008-9190-x. Google Scholar [18] G. Manno, On the geometry of Grassmannian equivalent connections,, Adv. Geom., 8 (2008), 329. doi: 10.1515/ADVGEOM.2008.021. Google Scholar [19] G. Manno, F. Oliveri and R. Vitolo, Differential equations uniquely determined by algebras of point symmetries,, Theoret. and Math. Phys., 151 (2007), 843. doi: 10.1007/s11232-007-0069-1. Google Scholar [20] L. Martina and P. Winternitz, Analysis and applications of the symmetry group of the multidimensional three-wave resonant interaction problem,, Ann. Physics, 196 (1989), 231. doi: 10.1016/0003-4916(89)90178-4. Google Scholar [21] M. A. McKiernan, On the convergence of series of iterates,, Publ. Math. Debrecen, 10 (1963), 30. Google Scholar [22] M. Mutz and D. Bensimon, Observation of toroidal vesicles,, Phys. Rev. A, 43 (1991), 4525. doi: 10.1103/PhysRevA.43.4525. Google Scholar [23] H. Naito, M. Okuda and O.-Y. Zhong-can, New Solutions to the Helfrich Variation Problem for the Shapes of Lipid Bilayer Vesicles: Beyond Delaunay's Surfaces,, Phys. Rev. Lett., 74 (1995), 4345. doi: 10.1103/PhysRevLett.74.4345. Google Scholar [24] D. Nelson, T. Piran and S. Weinberg, "Statistical Mechanics of Membranes and Surfaces,", World Scientific, (1989). Google Scholar [25] F. Neuman, Solution to the Problem No. 10 of N. Kamran,, in, (1985), 60. Google Scholar [26] P. J. Olver, "Applications of Lie Groups to Differential Equations,", Springer-Verlag, (1993). Google Scholar [27] L. V. Ovsiannikov, "Group Analysis of Differential Equations,", Academic Press, (1982). Google Scholar [28] L. Peliti, Amphiphilic Membranes,, in, (1994). Google Scholar [29] V. Pulov, M. Hadjilazova and I. M. Mladenov, Symmetries and Solutions of the Membrane Shape Equation,, talk given at, (2012). Google Scholar [30] R. Schmid, Infinite-dimensional Lie groups and algebras in mathematical physics,, Adv. Math. Phys., (2010). doi: 10.1155/2010/280362. Google Scholar [31] M. Schottenholer, "A Mathematical Introduction to Conformal Field Theory,", Lecture Notes in Physics, 759 (2008). Google Scholar [32] V. M. Vassilev, P. A. Djondjorov and I. M. Mladenov, Symmetry groups, conservation laws and group-invariant solutions of the membrane shape equation,, Geometry, (2006), 265. Google Scholar [33] V. M. Vassilev, P. A. Djondjorov and I. M. Mladenov, On the translationally-invariant solutions of the membrane shape equation,, Geometry, (2007), 312. Google Scholar [34] V. M. Vassilev, P. A. Djondjorov and I. M. Mladenov, Cylindrical equilibrium shapes of fluid membranes,, J. Phys. A, 41 (2008). doi: 10.1088/1751-8113/41/43/435201. Google Scholar [35] V. M. Vassiliev and I. M. Mladenov, Geometric symmetry groups, conservation laws and group-invariant solutions of the Willmore equation,, Geometry, (2004), 246. doi: 10.7546/giq-5-2004-246-265. Google Scholar [36] A. M. Vinogradov, Local symmetries and conservation laws,, Acta Appl. Math., 2 (1984), 21. doi: 10.1007/BF01405491. Google Scholar [37] T. J. Willmore, "Riemannian Geometry,", The Clarendon Press, (1993). Google Scholar [38] T. J. Willmore, "Total Curvature in Riemannian Geometry,", Ellis Horwood Ltd., (1982). Google Scholar [39] L. Weigu and M. Zhang, Embedding flows and smooth conjugacy,, Chinese Ann. Math. Ser. B, 18 (1997), 125. Google Scholar [40] P. Winternitz, Group Theory and exact solutions of partially integrable differential systems,, in, 310 (1990), 515. doi: 10.1007/978-94-009-0591-7_20. Google Scholar [41] M. Zhang, Embedding problem and functional equations,, Acta Math. Sinica (N.S.), 8 (1992), 148. doi: 10.1007/BF02629935. Google Scholar [42] W.-M. Zheng and J. Liu, The Helfrich equation for axisymmetric vesicles as a first integral,, Phys. Rev. E, 48 (1993), 2856. doi: 10.1103/PhysRevE.48.2856. Google Scholar [43] O.-Y. Zhong-can, Anchor ring-vesicle membranes,, Phys. Rev. A, 41 (1990), 4517. doi: 10.1103/PhysRevA.41.4517. Google Scholar [44] O.-Y. Zhong-can, Ji-Xing Liu and Yu-Zhang Xie, "Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases,", World Scientific, (1999). Google Scholar

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##### References:
 [1] W. A. Beyer and P. J. Channell, A functional equation for the embedding of a homeomorphism of the interval into a flow,, Lecture Notes in Math., 1163 (1985), 7. doi: 10.1007/BFb0076412. Google Scholar [2] L. Bianchi, "Lezioni di geometria differenziale,", Vol. 1, (1922). Google Scholar [3] J. Eells, The surfaces of Delaunay,, Math. Intelligencer, 9 (1987), 53. doi: 10.1007/BF03023575. Google Scholar [4] G. De Matteis, Group Analysis of the Membrane Shape Equation,, in, (2003), 221. doi: 10.1142/9789812704467_0031. Google Scholar [5] M. P. do Carmo, "Differential Geometry of Curves and Surfaces,", Prentice-Hall, (1976). Google Scholar [6] M. K. Fort Jr., The embedding of homeomorphisms in flows,, Proc. Amer. Math. Soc., 6 (1955), 960. doi: 10.1090/S0002-9939-1955-0080911-2. Google Scholar [7] R. Gilmore, "Lie Groups, Lie Algebras, and Some of Their Applications,", Robert E. Krieger Publishing Co., (1994). Google Scholar [8] M. Han, Conditions for a Diffeomorphism to be embedded in a $C^r$ flow,, Acta Math. Sinica (N.S.), 4 (1988), 111. doi: 10.1007/BF02560593. Google Scholar [9] W. Helfrich, Elastic properties of lipid bilayers: Theory and possible experiments,, Z. Naturforsch, 28c (1973), 693. Google Scholar [10] B. G. Konopelchenko, On solutions of the shape equation for membranes and strings,, Phys. Lett. B, 414 (1997), 58. doi: 10.1016/S0370-2693(97)01137-4. Google Scholar [11] H. Jian-Guo and O.-Y. Zhong-can, Shape equations of the axisymmetric vesicles,, Phys. Rev. E, 47 (1993), 461. doi: 10.1103/PhysRevE.47.461. Google Scholar [12] P. F. Lam, Embedding homeomorphisms in differential flows,, Colloq. Math., 35 (1976), 275. Google Scholar [13] P. F. Lam, Embedding a differentiable homeomorphism in a flow subject to a regularity condition on the derivatives of the positive transition homeomorphisms,, J. Differential Equations, 30 (1978), 31. doi: 10.1016/0022-0396(78)90021-9. Google Scholar [14] P. F. Lam, Embedding homeomorphisms in $C^1$-flows,, Ann. Mat. Pura Appl., 123 (1980), 11. doi: 10.1007/BF01796537. Google Scholar [15] R. A. Leo, L. Martina and G. Soliani, Group analysis of the three-wave resonant system in $(2+1)$-dimensions,, J. Math. Phys., 27 (1986), 2623. doi: 10.1063/1.527280. Google Scholar [16] R. Lipowsky and E. Sackman, "Structure and Dynamics of Membranes,", Elsevier Science B.V., (1995). Google Scholar [17] G. Manno and R. Vitolo, Geometric aspects of higher order variational principles on submanifolds,, Acta Appl. Math., 101 (2008), 215. doi: 10.1007/s10440-008-9190-x. Google Scholar [18] G. Manno, On the geometry of Grassmannian equivalent connections,, Adv. Geom., 8 (2008), 329. doi: 10.1515/ADVGEOM.2008.021. Google Scholar [19] G. Manno, F. Oliveri and R. Vitolo, Differential equations uniquely determined by algebras of point symmetries,, Theoret. and Math. Phys., 151 (2007), 843. doi: 10.1007/s11232-007-0069-1. Google Scholar [20] L. Martina and P. Winternitz, Analysis and applications of the symmetry group of the multidimensional three-wave resonant interaction problem,, Ann. Physics, 196 (1989), 231. doi: 10.1016/0003-4916(89)90178-4. Google Scholar [21] M. A. McKiernan, On the convergence of series of iterates,, Publ. Math. Debrecen, 10 (1963), 30. Google Scholar [22] M. Mutz and D. Bensimon, Observation of toroidal vesicles,, Phys. Rev. A, 43 (1991), 4525. doi: 10.1103/PhysRevA.43.4525. Google Scholar [23] H. Naito, M. Okuda and O.-Y. Zhong-can, New Solutions to the Helfrich Variation Problem for the Shapes of Lipid Bilayer Vesicles: Beyond Delaunay's Surfaces,, Phys. Rev. Lett., 74 (1995), 4345. doi: 10.1103/PhysRevLett.74.4345. Google Scholar [24] D. Nelson, T. Piran and S. Weinberg, "Statistical Mechanics of Membranes and Surfaces,", World Scientific, (1989). Google Scholar [25] F. Neuman, Solution to the Problem No. 10 of N. Kamran,, in, (1985), 60. Google Scholar [26] P. J. Olver, "Applications of Lie Groups to Differential Equations,", Springer-Verlag, (1993). Google Scholar [27] L. V. Ovsiannikov, "Group Analysis of Differential Equations,", Academic Press, (1982). Google Scholar [28] L. Peliti, Amphiphilic Membranes,, in, (1994). Google Scholar [29] V. Pulov, M. Hadjilazova and I. M. Mladenov, Symmetries and Solutions of the Membrane Shape Equation,, talk given at, (2012). Google Scholar [30] R. Schmid, Infinite-dimensional Lie groups and algebras in mathematical physics,, Adv. Math. Phys., (2010). doi: 10.1155/2010/280362. Google Scholar [31] M. Schottenholer, "A Mathematical Introduction to Conformal Field Theory,", Lecture Notes in Physics, 759 (2008). Google Scholar [32] V. M. Vassilev, P. A. Djondjorov and I. M. Mladenov, Symmetry groups, conservation laws and group-invariant solutions of the membrane shape equation,, Geometry, (2006), 265. Google Scholar [33] V. M. Vassilev, P. A. Djondjorov and I. M. Mladenov, On the translationally-invariant solutions of the membrane shape equation,, Geometry, (2007), 312. Google Scholar [34] V. M. Vassilev, P. A. Djondjorov and I. M. Mladenov, Cylindrical equilibrium shapes of fluid membranes,, J. Phys. A, 41 (2008). doi: 10.1088/1751-8113/41/43/435201. Google Scholar [35] V. M. Vassiliev and I. M. Mladenov, Geometric symmetry groups, conservation laws and group-invariant solutions of the Willmore equation,, Geometry, (2004), 246. doi: 10.7546/giq-5-2004-246-265. Google Scholar [36] A. M. Vinogradov, Local symmetries and conservation laws,, Acta Appl. Math., 2 (1984), 21. doi: 10.1007/BF01405491. Google Scholar [37] T. J. Willmore, "Riemannian Geometry,", The Clarendon Press, (1993). Google Scholar [38] T. J. Willmore, "Total Curvature in Riemannian Geometry,", Ellis Horwood Ltd., (1982). Google Scholar [39] L. Weigu and M. Zhang, Embedding flows and smooth conjugacy,, Chinese Ann. Math. Ser. B, 18 (1997), 125. Google Scholar [40] P. Winternitz, Group Theory and exact solutions of partially integrable differential systems,, in, 310 (1990), 515. doi: 10.1007/978-94-009-0591-7_20. Google Scholar [41] M. Zhang, Embedding problem and functional equations,, Acta Math. Sinica (N.S.), 8 (1992), 148. doi: 10.1007/BF02629935. Google Scholar [42] W.-M. Zheng and J. Liu, The Helfrich equation for axisymmetric vesicles as a first integral,, Phys. Rev. E, 48 (1993), 2856. doi: 10.1103/PhysRevE.48.2856. Google Scholar [43] O.-Y. Zhong-can, Anchor ring-vesicle membranes,, Phys. Rev. A, 41 (1990), 4517. doi: 10.1103/PhysRevA.41.4517. Google Scholar [44] O.-Y. Zhong-can, Ji-Xing Liu and Yu-Zhang Xie, "Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases,", World Scientific, (1999). Google Scholar
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