# 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 and Applied Analysis, 2014, 13 (1) : 453-481. doi: 10.3934/cpaa.2014.13.453
##### 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), 7-13. doi: 10.1007/BFb0076412. [2] L. Bianchi, "Lezioni di geometria differenziale," Vol. 1, Libraio Ed., Pisa, 1922. [3] J. Eells, The surfaces of Delaunay, Math. Intelligencer, 9 (1987), 53-57. doi: 10.1007/BF03023575. [4] G. De Matteis, Group Analysis of the Membrane Shape Equation, in "Nonlinear Physics: Theory and Experiment, II," World Scientific, River Edge NJ, (2003), 221-226. doi: 10.1142/9789812704467_0031. [5] M. P. do Carmo, "Differential Geometry of Curves and Surfaces," Prentice-Hall, Inc., Englewood Cliffs, NJ, 1976. [6] M. K. Fort Jr., The embedding of homeomorphisms in flows, Proc. Amer. Math. Soc., 6 (1955), 960-967. doi: 10.1090/S0002-9939-1955-0080911-2. [7] R. Gilmore, "Lie Groups, Lie Algebras, and Some of Their Applications," Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1994. [8] M. Han, Conditions for a Diffeomorphism to be embedded in a $C^r$ flow, Acta Math. Sinica (N.S.), 4 (1988), 111-123. doi: 10.1007/BF02560593. [9] W. Helfrich, Elastic properties of lipid bilayers: Theory and possible experiments, Z. Naturforsch, 28c (1973), 693-703. [10] B. G. Konopelchenko, On solutions of the shape equation for membranes and strings, Phys. Lett. B, 414 (1997), 58-64 . doi: 10.1016/S0370-2693(97)01137-4. [11] H. Jian-Guo and O.-Y. Zhong-can, Shape equations of the axisymmetric vesicles, Phys. Rev. E, 47 (1993), 461-467. doi: 10.1103/PhysRevE.47.461. [12] P. F. Lam, Embedding homeomorphisms in differential flows, Colloq. Math., 35 (1976), 275-287. [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), 31-40. doi: 10.1016/0022-0396(78)90021-9. [14] P. F. Lam, Embedding homeomorphisms in $C^1$-flows, Ann. Mat. Pura Appl., 123 (1980), 11-25. doi: 10.1007/BF01796537. [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), 2623-2628. doi: 10.1063/1.527280. [16] R. Lipowsky and E. Sackman, "Structure and Dynamics of Membranes," Elsevier Science B.V., Amsterdam, 1995. [17] G. Manno and R. Vitolo, Geometric aspects of higher order variational principles on submanifolds, Acta Appl. Math., 101 (2008), 215-229. doi: 10.1007/s10440-008-9190-x. [18] G. Manno, On the geometry of Grassmannian equivalent connections, Adv. Geom., 8 (2008), 329-342. doi: 10.1515/ADVGEOM.2008.021. [19] G. Manno, F. Oliveri and R. Vitolo, Differential equations uniquely determined by algebras of point symmetries, Theoret. and Math. Phys., 151 (2007), 843-850. doi: 10.1007/s11232-007-0069-1. [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), 231-277. doi: 10.1016/0003-4916(89)90178-4. [21] M. A. McKiernan, On the convergence of series of iterates, Publ. Math. Debrecen, 10 (1963), 30-39. [22] M. Mutz and D. Bensimon, Observation of toroidal vesicles, Phys. Rev. A, 43 (1991), 4525-4527. doi: 10.1103/PhysRevA.43.4525. [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), 4345-4348. doi: 10.1103/PhysRevLett.74.4345. [24] D. Nelson, T. Piran and S. Weinberg, "Statistical Mechanics of Membranes and Surfaces," World Scientific, Teaneck, NJ, 1989. [25] F. Neuman, Solution to the Problem No. 10 of N. Kamran, in "Proceedings, 23rd International Symposium on Functional Equations" (Gargnano, Italy) Centre for Information Theory, University of Waterloo, Ontario, Canada, (1985), 60-62. [26] P. J. Olver, "Applications of Lie Groups to Differential Equations," Springer-Verlag, New York, 1993. [27] L. V. Ovsiannikov, "Group Analysis of Differential Equations," Academic Press, New York-London, 1982. [28] L. Peliti, Amphiphilic Membranes, in "Fluctuating Geometries in Statistical Mechanics and Field Theory" (eds. F. David, P. Ginsparg and J. Zinn-Justin), Les Houches, (1994). [29] V. Pulov, M. Hadjilazova and I. M. Mladenov, Symmetries and Solutions of the Membrane Shape Equation, talk given at "XIV International Conference Geometry Integrability and Quantization'' (Varna, Bulgaria 2012), http://www.bio21.bas.bg/conference/Conference_files/sa12/slides/Pulov.pdf [30] R. Schmid, Infinite-dimensional Lie groups and algebras in mathematical physics, Adv. Math. Phys., (2010), Art ID 280362, 35 pp. doi: 10.1155/2010/280362. [31] M. Schottenholer, "A Mathematical Introduction to Conformal Field Theory," Lecture Notes in Physics, 759, Springer-Verlag, Berlin, 2008. [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, Integrability and Quantization, Softex, Sofia, (2006), 265-279. [33] V. M. Vassilev, P. A. Djondjorov and I. M. Mladenov, On the translationally-invariant solutions of the membrane shape equation, Geometry, Integrability and Quantization, Softex, Sofia, (2007), 312-321. [34] V. M. Vassilev, P. A. Djondjorov and I. M. Mladenov, Cylindrical equilibrium shapes of fluid membranes, J. Phys. A, 41 (2008), 435201, 16 pp. doi: 10.1088/1751-8113/41/43/435201. [35] V. M. Vassiliev and I. M. Mladenov, Geometric symmetry groups, conservation laws and group-invariant solutions of the Willmore equation, Geometry, Integrability and Quantization, Softex, Sofia, (2004), 246-265. doi: 10.7546/giq-5-2004-246-265. [36] A. M. Vinogradov, Local symmetries and conservation laws, Acta Appl. Math., 2 (1984), 21-78. doi: 10.1007/BF01405491. [37] T. J. Willmore, "Riemannian Geometry," The Clarendon Press, Oxford University Press, New York, 1993. [38] T. J. Willmore, "Total Curvature in Riemannian Geometry," Ellis Horwood Ltd., Chichester; Halsted Press, New York, 1982. [39] L. Weigu and M. Zhang, Embedding flows and smooth conjugacy, Chinese Ann. Math. Ser. B, 18 (1997), 125-138. [40] P. Winternitz, Group Theory and exact solutions of partially integrable differential systems, in "Partially integrable evolution equations in Physics," NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 310, Kluwer Acad. Publ., Dordrecht (1990), 515-567. doi: 10.1007/978-94-009-0591-7_20. [41] M. Zhang, Embedding problem and functional equations, Acta Math. Sinica (N.S.), 8 (1992), 148-157. doi: 10.1007/BF02629935. [42] W.-M. Zheng and J. Liu, The Helfrich equation for axisymmetric vesicles as a first integral, Phys. Rev. E, 48 (1993), 2856-2860. doi: 10.1103/PhysRevE.48.2856. [43] O.-Y. Zhong-can, Anchor ring-vesicle membranes, Phys. Rev. A, 41 (1990), 4517-4520. doi: 10.1103/PhysRevA.41.4517. [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, Hong Kong, 1999.

show all references

##### 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), 7-13. doi: 10.1007/BFb0076412. [2] L. Bianchi, "Lezioni di geometria differenziale," Vol. 1, Libraio Ed., Pisa, 1922. [3] J. Eells, The surfaces of Delaunay, Math. Intelligencer, 9 (1987), 53-57. doi: 10.1007/BF03023575. [4] G. De Matteis, Group Analysis of the Membrane Shape Equation, in "Nonlinear Physics: Theory and Experiment, II," World Scientific, River Edge NJ, (2003), 221-226. doi: 10.1142/9789812704467_0031. [5] M. P. do Carmo, "Differential Geometry of Curves and Surfaces," Prentice-Hall, Inc., Englewood Cliffs, NJ, 1976. [6] M. K. Fort Jr., The embedding of homeomorphisms in flows, Proc. Amer. Math. Soc., 6 (1955), 960-967. doi: 10.1090/S0002-9939-1955-0080911-2. [7] R. Gilmore, "Lie Groups, Lie Algebras, and Some of Their Applications," Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1994. [8] M. Han, Conditions for a Diffeomorphism to be embedded in a $C^r$ flow, Acta Math. Sinica (N.S.), 4 (1988), 111-123. doi: 10.1007/BF02560593. [9] W. Helfrich, Elastic properties of lipid bilayers: Theory and possible experiments, Z. Naturforsch, 28c (1973), 693-703. [10] B. G. Konopelchenko, On solutions of the shape equation for membranes and strings, Phys. Lett. B, 414 (1997), 58-64 . doi: 10.1016/S0370-2693(97)01137-4. [11] H. Jian-Guo and O.-Y. Zhong-can, Shape equations of the axisymmetric vesicles, Phys. Rev. E, 47 (1993), 461-467. doi: 10.1103/PhysRevE.47.461. [12] P. F. Lam, Embedding homeomorphisms in differential flows, Colloq. Math., 35 (1976), 275-287. [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), 31-40. doi: 10.1016/0022-0396(78)90021-9. [14] P. F. Lam, Embedding homeomorphisms in $C^1$-flows, Ann. Mat. Pura Appl., 123 (1980), 11-25. doi: 10.1007/BF01796537. [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), 2623-2628. doi: 10.1063/1.527280. [16] R. Lipowsky and E. Sackman, "Structure and Dynamics of Membranes," Elsevier Science B.V., Amsterdam, 1995. [17] G. Manno and R. Vitolo, Geometric aspects of higher order variational principles on submanifolds, Acta Appl. Math., 101 (2008), 215-229. doi: 10.1007/s10440-008-9190-x. [18] G. Manno, On the geometry of Grassmannian equivalent connections, Adv. Geom., 8 (2008), 329-342. doi: 10.1515/ADVGEOM.2008.021. [19] G. Manno, F. Oliveri and R. Vitolo, Differential equations uniquely determined by algebras of point symmetries, Theoret. and Math. Phys., 151 (2007), 843-850. doi: 10.1007/s11232-007-0069-1. [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), 231-277. doi: 10.1016/0003-4916(89)90178-4. [21] M. A. McKiernan, On the convergence of series of iterates, Publ. Math. Debrecen, 10 (1963), 30-39. [22] M. Mutz and D. Bensimon, Observation of toroidal vesicles, Phys. Rev. A, 43 (1991), 4525-4527. doi: 10.1103/PhysRevA.43.4525. [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), 4345-4348. doi: 10.1103/PhysRevLett.74.4345. [24] D. Nelson, T. Piran and S. Weinberg, "Statistical Mechanics of Membranes and Surfaces," World Scientific, Teaneck, NJ, 1989. [25] F. Neuman, Solution to the Problem No. 10 of N. Kamran, in "Proceedings, 23rd International Symposium on Functional Equations" (Gargnano, Italy) Centre for Information Theory, University of Waterloo, Ontario, Canada, (1985), 60-62. [26] P. J. Olver, "Applications of Lie Groups to Differential Equations," Springer-Verlag, New York, 1993. [27] L. V. Ovsiannikov, "Group Analysis of Differential Equations," Academic Press, New York-London, 1982. [28] L. Peliti, Amphiphilic Membranes, in "Fluctuating Geometries in Statistical Mechanics and Field Theory" (eds. F. David, P. Ginsparg and J. Zinn-Justin), Les Houches, (1994). [29] V. Pulov, M. Hadjilazova and I. M. Mladenov, Symmetries and Solutions of the Membrane Shape Equation, talk given at "XIV International Conference Geometry Integrability and Quantization'' (Varna, Bulgaria 2012), http://www.bio21.bas.bg/conference/Conference_files/sa12/slides/Pulov.pdf [30] R. Schmid, Infinite-dimensional Lie groups and algebras in mathematical physics, Adv. Math. Phys., (2010), Art ID 280362, 35 pp. doi: 10.1155/2010/280362. [31] M. Schottenholer, "A Mathematical Introduction to Conformal Field Theory," Lecture Notes in Physics, 759, Springer-Verlag, Berlin, 2008. [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, Integrability and Quantization, Softex, Sofia, (2006), 265-279. [33] V. M. Vassilev, P. A. Djondjorov and I. M. Mladenov, On the translationally-invariant solutions of the membrane shape equation, Geometry, Integrability and Quantization, Softex, Sofia, (2007), 312-321. [34] V. M. Vassilev, P. A. Djondjorov and I. M. Mladenov, Cylindrical equilibrium shapes of fluid membranes, J. Phys. A, 41 (2008), 435201, 16 pp. doi: 10.1088/1751-8113/41/43/435201. [35] V. M. Vassiliev and I. M. Mladenov, Geometric symmetry groups, conservation laws and group-invariant solutions of the Willmore equation, Geometry, Integrability and Quantization, Softex, Sofia, (2004), 246-265. doi: 10.7546/giq-5-2004-246-265. [36] A. M. Vinogradov, Local symmetries and conservation laws, Acta Appl. Math., 2 (1984), 21-78. doi: 10.1007/BF01405491. [37] T. J. Willmore, "Riemannian Geometry," The Clarendon Press, Oxford University Press, New York, 1993. [38] T. J. Willmore, "Total Curvature in Riemannian Geometry," Ellis Horwood Ltd., Chichester; Halsted Press, New York, 1982. [39] L. Weigu and M. Zhang, Embedding flows and smooth conjugacy, Chinese Ann. Math. Ser. B, 18 (1997), 125-138. [40] P. Winternitz, Group Theory and exact solutions of partially integrable differential systems, in "Partially integrable evolution equations in Physics," NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 310, Kluwer Acad. Publ., Dordrecht (1990), 515-567. doi: 10.1007/978-94-009-0591-7_20. [41] M. Zhang, Embedding problem and functional equations, Acta Math. Sinica (N.S.), 8 (1992), 148-157. doi: 10.1007/BF02629935. [42] W.-M. Zheng and J. Liu, The Helfrich equation for axisymmetric vesicles as a first integral, Phys. Rev. E, 48 (1993), 2856-2860. doi: 10.1103/PhysRevE.48.2856. [43] O.-Y. Zhong-can, Anchor ring-vesicle membranes, Phys. Rev. A, 41 (1990), 4517-4520. doi: 10.1103/PhysRevA.41.4517. [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, Hong Kong, 1999.
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