The Geometry of Fluid Membranes: Variational Principles, Symmetries and Conservation Laws

  • Jemal GuvenEmail author
  • Pablo Vázquez-Montejo
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 577)


The behavior of a lipid membrane on mesoscopic scales is captured unusually accurately by its geometrical degrees of freedom. Indeed, the membrane geometry is, very often, a direct reflection of the physical state of the membrane. In this chapter we will examine the intimate connection between the geometry and the physics of fluid membranes from a number of points of view. We begin with a review of the description of the surface geometry in terms of the metric and the extrinsic curvature, examining surface deformations in terms of the behavior of these two tensors. The shape equation describing membrane equilibrium is derived and the qualitative behavior of solutions described. We next look at the conservation laws implied by the Euclidean invariance of the energy, describing the remarkably simple relationship between the stress distributed in the membrane and its geometry. This relationship is used to examine membrane-mediated interactions. We show how this geometrical framework can be extended to accommodate constraints—both global and local—as well as additional material degrees of freedom coupling to the geometry. The conservation laws are applied to examine the response of an axially symmetric membrane to localized external forces and to characterize topologically nontrivial states. We wrap up by looking at the conformal invariance of the symmetric two-dimensional bending energy, and examine some of its consequences.


Height Function Extrinsic Curvature Spontaneous Curvature Fundamental Tensor Shape Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



JG would like to thank David Steigmann for the invitation to lecture at the CISM Summer School held in Udine, Italy during July of 2016. This chapter is based on these lectures. We would like also to thank Markus Deserno, Martin Müller and Saša Svetina for their valuable input. This work was partially supported by CONACyT grant 180901.


  1. L. Amoasii, K. Hnia, G. Chicanne, A. Brech, B.S. Cowling, M.M. Müller, Y. Schwab, P. Koebel, A. Ferry, B. Payrastre, J. Laporte, Myotubularin and ptdins3p remodel the sarcoplasmic reticulum in muscle in vivo. J. Cell Sci. 126(8), 1806–1819 (2013). doi: 10.1242/jcs.118505 CrossRefGoogle Scholar
  2. R. Arnowitt, S. Deser, C.W. Misner, Dynamical structure and definition of energy in general relativity. Phys. Rev. 116(5), 1322–1330 (1959). doi: 10.1103/PhysRev.116.1322
  3. G. Arreaga, R. Capovilla, J. Guven, Noether currents for bosonic branes. Ann. Phys. 279(1), 126–158 (2000). doi: 10.1006/aphy.1999.5979 MathSciNetCrossRefzbMATHGoogle Scholar
  4. M. Arroyo, A. DeSimone, Relaxation dynamics of fluid membranes. Phys. Rev. E 79(3), 031915 (2009). doi: 10.1103/PhysRevE.79.031915
  5. P. Bassereau, B. Sorre, A. Lévy, Bending lipid membranes: experiments after w. helfrich’s model. Adv. Colloid Interface Sci. 208, 47–57 (2014). doi: 10.1016/j.cis.2014.02.002. Special issue in honour of Wolfgang Helfrich
  6. Y. Bernard, Noether’s theorem and the willmore functional. Adv. Calc. Var. (2015). doi: 10.1515/acv-2014-0033 zbMATHGoogle Scholar
  7. L. Bouzar, F. Menas, M.M. Müller, Toroidal membrane vesicles in spherical confinement. Phys. Rev. E 92, 032721 (2015). doi: 10.1103/PhysRevE.92.032721 CrossRefGoogle Scholar
  8. B. Božič, J. Guven, P. Vázquez-Montejo, S. Svetina, Direct and remote constriction of membrane necks. Phys. Rev. E 89, 052701 (2014). doi: 10.1103/PhysRevE.89.052701 Google Scholar
  9. B. Božič, S.L. Das, S. Svetina, Sorting of integral membrane proteins mediated by curvature-dependent protein-lipid bilayer interaction. Soft Matter 11, 2479–2487 (2015). doi: 10.1039/C4SM02289K CrossRefGoogle Scholar
  10. P.B. Canham, The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. J. Theor. Biol. 26(1), 61–81 (1970). doi: 10.1016/S0022-5193(70)80032-7 CrossRefGoogle Scholar
  11. R. Capovilla, J. Guven, Geometry of lipid vesicle adhesion. Phys. Rev. E 66, 041604 (2002a). doi: 10.1103/PhysRevE.66.041604 CrossRefGoogle Scholar
  12. R. Capovilla, J. Guven, Stresses in lipid membranes. J. Phys. A Math. Gen. 35(30), 6233 (2002b). doi: 10.1088/0305-4470/35/30/302
  13. R. Capovilla, J. Guven, Stress and geometry of lipid vesicles. J. Phys.-Condens. Mat. 16, S2187–S2191 (2004a). doi: 10.1088/0953-8984/16/22/018
  14. R. Capovilla, J. Guven, Second variation of the Helfrich-Canham Hamiltonian and reparametrization invariance. J. Phys. A Math. Gen. 37(23), 5983 (2004b). doi: 10.1088/0305-4470/37/23/003 MathSciNetCrossRefzbMATHGoogle Scholar
  15. R. Capovilla, J. Guven, J.A. Santiago, Lipid membranes with an edge. Phys. Rev. E 66, 021607 (2002). doi: 10.1103/PhysRevE.66.021607 CrossRefGoogle Scholar
  16. R. Capovilla, J. Guven, J.A. Santiago, Deformations of the geometry of lipid vesicles. J. Phys. A Math. Gen. 36(23), 6281 (2003). doi: 10.1088/0305-4470/36/23/301
  17. P. Castro-Villarreal, J. Guven, Axially symmetric membranes with polar tethers. J. Phys. A Math. Theor. 40(16), 4273 (2007a). doi: 10.1088/1751-8113/40/16/002 MathSciNetCrossRefzbMATHGoogle Scholar
  18. P. Castro-Villarreal, J. Guven, Inverted catenoid as a fluid membrane with two points pulled together. Phys. Rev. E 76, 011922 (2007b). doi: 10.1103/PhysRevE.76.011922 CrossRefGoogle Scholar
  19. M. Deserno, Membrane elasticity and mediated interactions in continuum theory: a differential geometric approach, in Biomembrane Frontiers, ed. by R. Faller, M.L. Longo, S.H. Risbud, T. Jue. Handbook of Modern Biophysics (Humana Press, New York, 2009), pp. 41–74. doi: 10.1007/978-1-60761-314-5_2
  20. M. Deserno, Fluid lipid membranes: from differential geometry to curvature stresses. Chem. Phys. Lipids 185, 11–45 (2015). doi: 10.1016/j.chemphyslip.2014.05.001. Membrane mechanochemistry: From the molecular to the cellular scale
  21. M. Deserno, M.M. Müller, J. Guven, Contact lines for fluid surface adhesion. Phys. Rev. E 76, 011605 (2007). doi: 10.1103/PhysRevE.76.011605 CrossRefGoogle Scholar
  22. P. Diggins IV, Z.A. McDargh, M. Deserno, Curvature softening and negative compressibility of gel-phase lipid membranes. J. Am. Chem. Soc. 137(40), 12752–12755 (2015). doi: 10.1021/jacs.5b06800 CrossRefGoogle Scholar
  23. M. Do Carmo, Differential Geometry of Curves and Surface (Prentice Hall, Upper Saddle River, 1976)Google Scholar
  24. M. Do Carmo. Riemannian Geometry. (Birkhauser, Basel, 1992)Google Scholar
  25. P.G. Dommersnes, J.-B. Fournier, The many-body problem for anisotropic membrane inclusions and the self-assembly of saddle defects into an egg carton. Biophys. J. 83, 2898–2905 (2002). doi: 10.1016/S0006-3495(02)75299-5 CrossRefGoogle Scholar
  26. E.A. Evans, Bending resistance and chemically induced moments in membrane bilayers. Biophys. J. 14, 923–931 (1974). doi: 10.1016/S0006-3495(74)85959-X CrossRefGoogle Scholar
  27. E.A. Evans, R. Skalak, Mechanics and Thermodynamics of Biomembranes (CRC Press, Boca Raton, 1980)Google Scholar
  28. J.-B. Fournier, On the stress and torque tensors in fluid membranes. Soft Matter 3, 883–888 (2007). doi: 10.1039/B701952A CrossRefGoogle Scholar
  29. J.-B. Fournier, Dynamics of the force exchanged between membrane inclusions. Phys. Rev. Lett. 112, 128101 (2014). doi: 10.1103/PhysRevLett.112.128101 CrossRefGoogle Scholar
  30. J.-B. Fournier, P. Galatola, High-order power series expansion of the elastic interaction between conical membrane inclusions. Eur. Phys. J. E 38(8) (2015). doi: 10.1140/epje/i2015-15086-3
  31. R. Goetz, W. Helfrich, The egg carton: theory of a periodic superstructure of some lipid membranes. J. Phys. II Fr. 6(2), 215–223 (1996). doi: 10.1051/jp2:1996178 Google Scholar
  32. M. Goulian, R. Bruinsma, P. Pincus, Long-range forces in heterogeneous fluid membranes. EPL (Europhysics Letters) 22(2), 145 (1993). doi: 10.1209/0295-5075/22/2/012 CrossRefGoogle Scholar
  33. J. Guven, Membrane geometry with auxiliary variables and quadratic constraints. J. Phys. A Math. Gen. 37(28), L313 (2004). doi: 10.1088/0305-4470/37/28/L02
  34. J. Guven, Conformally invariant bending energy for hypersurfaces. J. Phys. A Math. Gen. 38(37), 7943 (2005). doi: 10.1088/0305-4470/38/37/002
  35. J. Guven, Laplace pressure as a surface stress in fluid vesicles. J. Phys. A Math. Gen. 39(14), 3771 (2006). doi: 10.1088/0305-4470/39/14/019
  36. J. Guven, M.M. Müller, How paper folds: bending with local constraints. J. Phys. A Math. Theo. 41(5), 055203 (2008). doi: 10.1088/1751-8113/41/5/055203
  37. J. Guven, M.M. Müller, P. Vázquez-Montejo, Conical instabilities on paper. J. Phys. A Math. Theo. 45(1), 015203 (2012). doi: 10.1088/1751-8113
  38. J. Guven, P. Vázquez-Montejo, Spinor representation of surfaces and complex stresses on membranes and interfaces. Phys. Rev. E 82, 041604 (2010). doi: 10.1103/PhysRevE.82.041604 MathSciNetCrossRefGoogle Scholar
  39. J. Guven, P. Vázquez-Montejo, Constrained metric variations and emergent equilibrium surfaces. Phys. Lett. A 377(23–24), 1507–1511 (2013a). doi: 10.1016/j.physleta.2013.04.031
  40. J. Guven, P. Vázquez-Montejo, Force dipoles and stable local defects on fluid vesicles. Phys. Rev. E 87, 042710 (2013b). doi: 10.1103/PhysRevE.87.042710 CrossRefGoogle Scholar
  41. J. Guven, G. Huber, D.M. Valencia, Terasaki spiral ramps in the rough endoplasmic reticulum. Phys. Rev. Lett. 113, 188101 (2014). doi: 10.1103/PhysRevLett.113.188101 CrossRefGoogle Scholar
  42. R.C. Haussman, M. Deserno, Effective field theory of thermal casimir interactions between anisotropic particles. Phys. Rev. E 89, 062102 (2014). doi: 10.1103/PhysRevE.89.062102 CrossRefGoogle Scholar
  43. W. Helfrich, Elastic properties of lipid bilayers, theory and possible experiments. Z. Naturforsch. C 28, 693–703 (1973).
  44. J.H. Jellett. Sur la surface dont la courbure moyenne est constante. Journal de Mathematiques Pures et Appliquees, 163–167 (1853)Google Scholar
  45. J.T. Jenkins, The equations of mechanical equilibrium of a model membrane. SIAM J. Appl. Math. 32(4), 755–764 (1977). doi: 10.1137/0132063 MathSciNetCrossRefzbMATHGoogle Scholar
  46. F. Jülicher, The morphology of vesicles of higher topological genus: conformal degeneracy and conformal modes. J. Phys. II Fr. 6(12), 1797–1824 (1996). doi: 10.1051/jp2:1996161 Google Scholar
  47. F. Jülicher, U. Seifert, Shape equations for axisymmetric vesicles: a clarification. Phys. Rev. E 49, 4728–4731 (1994). doi: 10.1103/PhysRevE.49.4728 CrossRefGoogle Scholar
  48. F. Jülicher, U. Seifert, R. Lipowsky, Conformal degeneracy and conformal diffusion of vesicles. Phys. Rev. Lett. 71, 452–455 (1993). doi: 10.1103/PhysRevLett.71.452 CrossRefGoogle Scholar
  49. O. Kahraman, N. Stoop, M.M. Müller, Morphogenesis of membrane invaginations in spherical confinement. EPL (Europhysics Letters) 97(6), 68008 (2012a). doi: 10.1209/0295-5075/97/68008 CrossRefGoogle Scholar
  50. O. Kahraman, N. Stoop, M.M. Müller, Fluid membrane vesicles in confinement. New J. Phys. 14(9), 095021 (2012b). doi: 10.1088/1367-2630/14/9/095021 CrossRefGoogle Scholar
  51. K.S. Kim, J. Neu, G. Oster, Curvature-mediated interactions between membrane proteins. Biophys. J. 75(5), 2274–2291 (1998). doi: 10.1016/S0006-3495(98)77672-6 CrossRefGoogle Scholar
  52. M.M. Kozlov, Fission of biological membranes: interplay between dynamin and lipids. Traffic 2(1), 51–65 (2001). doi: 10.1034/j.1600-0854.2001.020107.x CrossRefGoogle Scholar
  53. V. Kralj-Iglič, S. Svetina, B. Žekž, Shapes of bilayer vesicles with membrane embedded molecules. Eur. Biophys. J. 24(5), 311–321 (1996). doi: 10.1007/BF00180372 CrossRefGoogle Scholar
  54. V. Kralj-Iglič, V. Heinrich, S. Svetina, B. Žekž, Free energy of closed membrane with anisotropic inclusions. Eur. Phys. J. B - Condens. Matter Complex Syst. 10(1), 5–8 (1999). doi: 10.1007/s100510050822
  55. E. Kreyszig, Differential Geometry (Dover Publications, New York, 1991)zbMATHGoogle Scholar
  56. R. Kusner, Geometric analysis and computer graphics, in Mathematical Sciences Research Institute Publications, vol. 17, ed. by P. Concus, R. Finn, D.A. Hoffman (Springer, New York, 1991), pp. 103–108. doi: 10.1007/978-1-4613-9711-3_11
  57. R. Lipowsky, Spontaneous tubulation of membranes and vesicles reveals membrane tension generated by spontaneous curvature. Faraday Discuss. 161, 305–331 (2013). doi: 10.1039/C2FD20105D CrossRefGoogle Scholar
  58. M.A. Lomholt, L. Miao, Descriptions of membrane mechanics from microscopic and effective two-dimensional perspectives. J. Phys. A Math. Gen. 39(33), 10323 (2006). doi: 10.1088/0305-4470/39/33/005
  59. O.V. Manyuhina, J.J. Hetzel, M.I. Katsnelson, A. Fasolino, Non-spherical shapes of capsules within a fourth-order curvature model. Eur. Phys. J. E 32(3), 223–228 (2010). doi: 10.1140/epje/i2010-10631-2 CrossRefGoogle Scholar
  60. F.C. Marques, A. Neves, Min-Max theory and the Willmore conjecture. Ann. Math. Second Series 179(2), 683–782 (2014a). doi: 10.4007/annals.2014.179.2.6
  61. F.C. Marques, A. Neves, The Willmore conjecture. Jahresbericht der Deutschen Mathematiker-Vereinigung 116(4), 201–222 (2014b). doi: 10.1365/s13291-014-0104-8
  62. Z. McDargh, P. Vázquez-Montejo, J. Guven, M. Deserno. Constriction by dynamin: Elasticity vs. adhesion. Biophy. J. 111(11), 2470–2480 (2016). doi: 10.1016/j.bpj.2016.10.019
  63. X. Michalet, D. Bensimon, Observation of stable shapes and conformal diffusion in genus 2 vesicles. Science 269(5224), 666–668 (1995). doi: 10.1126/science.269.5224.666 CrossRefGoogle Scholar
  64. S. Morlot, A. Roux, Mechanics of dynamin-mediated membrane fission. Ann. Rev. Biophys. 42(1), 629–649 (2013). doi: 10.1146/annurev-biophys-050511-102247 CrossRefGoogle Scholar
  65. M.M. Müller, Theoretical studies of fluid membrane mechanics, Ph.D. thesis, University of Mainz (Germany), 2007Google Scholar
  66. M.M. Müller, M. Deserno, J. Guven, Geometry of surface-mediated interactions. Europhys. Lett. 69(3), 482 (2005a). doi: 10.1209/epl/i2004-10368-1 CrossRefGoogle Scholar
  67. M.M. Müller, M. Deserno, J. Guven, Interface-mediated interactions between particles: a geometrical approach. Phys. Rev. E 72, 061407 (2005b). doi: 10.1103/PhysRevE.72.061407 MathSciNetCrossRefGoogle Scholar
  68. M.M. Müller, M. Deserno, J. Guven, Balancing torques in membrane-mediated interactions: exact results and numerical illustrations. Phys. Rev. E 76, 011921 (2007). doi: 10.1103/PhysRevE.76.011921 CrossRefGoogle Scholar
  69. M. Mutz, D. Bensimon, Observation of toroidal vesicles. Phys. Rev. A 43, 4525–4527 (1991). doi: 10.1103/PhysRevA.43.4525 CrossRefGoogle Scholar
  70. G.-M. Nam, N.-K. Lee, H. Mohrbach, A. Johner, I.M. Kulić, Helices at interfaces. EPL (Europhysics Letters) 100(2), 28001 (2012). doi: 10.1209/0295-5075/100/28001 CrossRefGoogle Scholar
  71. H. Noguchi, Construction of nuclear envelope shape by a high-genus vesicle with pore-size constraint. Biophy. J. 111(4), 824–831 (2016a). doi: 10.1016/j.bpj.2016.07.010
  72. H. Noguchi, Membrane tubule formation by banana-shaped proteins with or without transient network structur. Sci. Rep. 6, 20935 (2016b). doi: 10.1038/srep20935
  73. A.S.H. Noguchi, M. Imai, Shape transformations of toroidal vesicles. Soft Matter 11, 193–201 (2015)CrossRefGoogle Scholar
  74. Z.-C. Ou-Yang, Anchor ring-vesicle membranes. Phys. Rev. A 41, 4517–4520 (1990). doi: 10.1103/PhysRevA.41.4517 CrossRefGoogle Scholar
  75. Z.-C. Ou-Yang, W. Helfrich, Instability and deformation of a spherical vesicle by pressure. Phys. Rev. Lett. 59, 2486–2488 (1987). doi: 10.1103/PhysRevLett.59.2486 CrossRefGoogle Scholar
  76. Z.-C. Ou-Yang, W. Helfrich, Bending energy of vesicle membranes: General expressions for the first, second, and third variation of the shape energy and applications to spheres and cylinders. Phys. Rev. A 39, 5280–5288 (1989). doi: 10.1103/PhysRevA.39.5280 CrossRefGoogle Scholar
  77. Z.C. Ou-Yang, J.X. Liu, Y.Z. Xie, X. Yu-Zhang, Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases, Advanced series on theoretical physical science (World Scientific, Singapore, 1999)zbMATHGoogle Scholar
  78. R. Phillips, T. Ursell, P. Wiggins, P. Sens, Emerging roles for lipids in shaping membrane-protein function. Nature 459, 379–385 (2009). doi: 10.1038/nature08147 CrossRefGoogle Scholar
  79. U. Pinkall, Cyclides of Dupin, in Mathematical Models from the Collections of Universities and Museums, ed. by E.G. Fischer. Advanced Lectures in Mathematics Series (Friedrick Vieweg & Son, Braunschweig, 1986), pp. 28–30. Chap. 3.3Google Scholar
  80. R. Podgornik, S. Svetina, B. Žekš, Parametrization invariance and shape equations of elastic axisymmetric vesicles. Phys. Rev. E 51, 544–547 (1995). doi: 10.1103/PhysRevE.51.544 CrossRefGoogle Scholar
  81. T.R. Powers, Dynamics of filaments and membranes in a viscous fluid. Rev. Mod. Phy. 82(2), 1607–1631 (2010). doi: 10.1103/RevMod-Phys.82.1607
  82. B.J. Reynwar, G. Illya, V.A. Harmandaris, M.M. Müller, K. Kremer, M. Deserno, Aggregation and vesiculation of membrane proteins by curvature-mediated interactions. Nature 447, 461–464 (2007). doi: 10.1038/nature05840 CrossRefGoogle Scholar
  83. Y. Schweitzer, M. Kozlov, Membrane-mediated interaction between strongly anisotropic protein scaffolds. PLoS Comput. Biol. 11, 1004054 (2015). doi: 10.1371/journal.pcbi.1004054 CrossRefGoogle Scholar
  84. U. Seifert, Conformal transformations of vesicle shapes. J. Phys. A Math. Gen. 24(11), 573 (1991). doi: 10.1088/0305-4470/24/11/001
  85. U. Seifert, Vesicles of toroidal topology. Phys. Rev. Lett. 66, 2404–2407 (1991). doi: 10.1103/PhysRevLett.66.2404 CrossRefGoogle Scholar
  86. U. Seifert, Configurations of fluid membranes and vesicles. Adv. Phys. 46(1), 13–137 (1997). doi: 10.1080/00018739700101488 CrossRefGoogle Scholar
  87. U. Seifert, R. Lipowsky, Morphology of vesicles, in Structure and Dynamics of Membranes From Cells to Vesicles, ed. by R. Lipowsky, E. Sackmann. Handbook of Biological Physics, vol. 1 (North-Holland, Amsterdam, 1995), pp. 403–463. doi: 10.1016/S1383-8121(06)80025-4
  88. P. Sens, L. Johannes, P. Bassereau, Biophysical approaches to protein-induced membrane deformations in trafficking. Current Opinion Cell Biol. 20(4), 476–482 (2008). doi: 10.1016/ CrossRefGoogle Scholar
  89. H. Shiba, H. Noguchi, J.-B. Fournier, Monte carlo study of the frame, fluctuation and internal tensions of fluctuating membranes with fixed area. Soft Matter 12, 2373–2380 (2016). doi: 10.1039/C5SM01900A
  90. M. Spivak, A Comprehensive Introduction to Differential Geometry, vol. 1–5, 3rd edn. (Publish or Perish, Inc., Houston, 1999)Google Scholar
  91. D.J. Steigmann, Fluid films with curvature elasticity. Arch. Rational Mech. Anal. 150(2), 127–152 (1999). doi: 10.1007/s002050050183 MathSciNetCrossRefzbMATHGoogle Scholar
  92. S. Svetina, B. Žekž, Membrane bending energy and shape determination of phospholipid vesicles and red blood cells. Eur. Biophys. J. 17(2), 101–111 (1989). doi: 10.1007/BF00257107 CrossRefGoogle Scholar
  93. S. Svetina, B. Žekš, Nonlocal membrane bending: a reflection, the facts and its relevance. Adv. Colloid Interface Sci. 208, 189–196 (2014). doi: 10.1016/j.cis.2014.01.010. Special issue in honour of Wolfgang Helfrich
  94. M. Terasaki, T. Shemesh, N. Kasthuri, R.W. Klemm, R. Schalek, K.J. Hayworth, A.R. Hand, M. Yankova, G. Huber, J.W. Lichtman, T.A. Rapoport, M.M. Kozlov, Stacked endoplasmic reticulum sheets are connected by helicoidal membrane motifs. Cell 154, 285–296 (2013). doi: 10.1016/j.cell.2013.06.031 CrossRefGoogle Scholar
  95. Z.C. Tu, Z.C. Ou-Yang, Lipid membranes with free edges. Phys. Rev. E 68, 061915 (2003). doi: 10.1103/PhysRevE.68.061915 CrossRefGoogle Scholar
  96. Z.C. Tu, Z.C. Ou-Yang, A geometric theory on the elasticity of bio-membranes. J. Phys. A Math. Gen. 37(47), 11407 (2004). doi: 10.1088/0305-4470/37/47/010
  97. Z.C. Tu, Z.C. Ou-Yang, Recent theoretical advances in elasticity of membranes following helfrich’s spontaneous curvature model. Adv. Colloid Interface Sci. 208, 66–75 (2014). doi: 10.1016/j.cis.2014.01.008. Special issue in honour of Wolfgang Helfrich
  98. R.M. Wald, General Relativity (University of Chicago Press, Chicago, 2010)zbMATHGoogle Scholar
  99. T.R. Weikl, M.M. Kozlov, W. Helfrich, Interaction of conical membrane inclusions: effect of lateral tension. Phys. Rev. E 57, 6988–6995 (1998). doi: 10.1103/PhysRevE.57.6988 CrossRefGoogle Scholar
  100. T.J. Willmore, Note on embedded surfaces. An. St. Univ. Iasi, sIa Mat. B 12, 493–496 (1965)Google Scholar
  101. T.J. Willmore, Total Curvature in Riemannian Geometry (Ellis Horwood, Chichester, 1982)zbMATHGoogle Scholar
  102. T.J. Willmore, Riemannian Geometry (Oxford University Press, Oxford, 1996)zbMATHGoogle Scholar
  103. C. Yolcu, M. Deserno, Membrane-mediated interactions between rigid inclusions: an effective field theory. Phys. Rev. E 86, 031906 (2012). doi: 10.1103/PhysRevE.86.031906 CrossRefGoogle Scholar
  104. C. Yolcu, I.Z. Rothstein, M. Deserno, Effective field theory approach to casimir interactions on soft matter surfaces. EPL (Europhysics Letters) 96(2), 20003 (2011). doi: 10.1209/0295-5075/96/20003 CrossRefGoogle Scholar
  105. C. Yolcu, I.Z. Rothstein, M. Deserno, Effective field theory approach to fluctuation-induced forces between colloids at an interface. Phys. Rev. E 85, 011140 (2012). doi: 10.1103/PhysRevE.85.011140 CrossRefGoogle Scholar
  106. C. Yolcu, R.C. Haussman, M. Deserno, The effective field theory approach towards membrane-mediated interactions between particles. Adv. Colloid Interface Sci. 208, 89–109 (2014). doi: 10.1016/j.cis.2014.02.017. Special issue in honour of Wolfgang Helfrich
  107. W.-M. Zheng, J. Liu, Helfrich shape equation for axisymmetric vesicles as a first integral. Phys. Rev. E 48, 2856–2860 (1993). doi: 10.1103/PhysRevE.48.2856 CrossRefGoogle Scholar

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© CISM International Centre for Mechanical Sciences 2018

Authors and Affiliations

  1. 1.Instituto de Ciencias NuclearesUniversidad Nacional Autónoma de MéxicoMexico CityMexico
  2. 2.Facultad de MatemáticasUniversidad Autónoma de YucatánMéridaMexico

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