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

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

Abstract

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.

Keywords

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.

Notes

Acknowledgements

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.

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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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