Geometry of the anisotropic minimal surfaces

Ivaïlo M. Mladenov; Mariana Ts. Hadzhilazova

doi:10.2478/v10309-012-0042-3

.blurhash-client-img { display: none !important; }

Geometry of the anisotropic minimal surfaces

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

Volume 20 (2012): Issue 2 (June 2012)

By: Ivaïlo M. Mladenov and Mariana Ts. Hadzhilazova

Open Access

|May 2013

Abstract

A simple modification of the surface tension in the axisymmetric case leads to analogues of the Delaunay surfaces. Here we have derived an explicit parameterization of the most simple case of this new class of surfaces which can be considered as a generalization of the catenoids. The geometry of these surfaces depends on two real parameters and has been studied in some detail

References

[1] Blyth M. and Pozrikidis C., Solution Space of Axisymmetric Capsules Enclosed by Elastic Membranes, Eur. J. Mech. A 23 (2004) 877-892.10.1016/j.euromechsol.2004.05.007
Search in Google Scholar
[2] Chen B. and Kamien R., Nematic Films and Radially Anisotropic Delau- nay Surfaces, Eur. Phys. J. E 28 (2009) 315-329.10.1140/epje/i2008-10441-1
Search in Google Scholar
[3] Delaunay C., Sur la surface de revolution dont la courbure moyenne est constante, J. Math. Pures et Appliquées, 6 (1841) 309-320.
Search in Google Scholar
[4] Djondjorov P., Hadzhilazova M., Mladenov I. and Vassilev V., Beyond Delaunay Surfaces, J. Geom. Symmetry Phys. 18 (2010) 1-11.
Search in Google Scholar
[5] Hu J.-G. and Ou-Yang Z.-C., Shape Equations of the Axisymmetric Vesi- cles, Phys. Rev. E 47 (1993) 461-467.10.1103/PhysRevE.47.461
Search in Google Scholar
[6] Konopelchenko B., On Solutions of the Shape Equation for Membranes and Strings, Phys. Lett. B 414 (1997) 58-64.10.1016/S0370-2693(97)01137-4
Search in Google Scholar
[7] Lipowsky R. and Sackmann E. (Eds), Handbook of Biological Physics (Structure and Dynamics of Membranes) vol 1, Elsevier, Amsterdam 1995.10.1016/S1383-8121(06)80018-7
Search in Google Scholar
[8] Ludu A., Nonlinear Waves and Solitons on Contours and Closed Surfaces, Springer, Berlin 2007.
Search in Google Scholar
[9] Mladenov I., New Solutions of the Shape Equation, Eur. Phys. J. B 29 (2002) 327-330.10.1140/epjb/e2002-00310-y
Search in Google Scholar
[10] Mladenov I., Hadzhilazova M., Djondjorov P. and Vassilev V., Some Explicit Solutions of the Shape Equation, In: AIP Conference Proceedings 1301 (2010) 215-227.10.1063/1.3526617
Search in Google Scholar
[11] Nitsche J., Boundary Value Problems for Variational Integrals Involving Surface Curvatures, Quarterly Appl. Math. 60 (1993) 363-387.10.1090/qam/1218374
Search in Google Scholar
[12] Oprea J., Di erential Geometry and Its Applications, Mathematical Association of America, Washington D. C. 2007.
Search in Google Scholar
[13] Ou-Yang Z.-C. and Helfrich W., 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 (1989) 5280-5288.10.1103/PhysRevA.39.5280
Search in Google Scholar
[14] Ou-Yang Z.-C., Selection of Toroidal Shape of Partially Polymerized Membranes, Phys. Rev. E 47 (1993) 747-749.10.1103/PhysRevE.47.7479960058
Search in Google Scholar
[15] Ou-Yang Z.-C., Liu J.-X., and Xie Y.-Z., Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases, World Scientific, Hong Kong 1999.
Search in Google Scholar
[16] Vassilev V., Djondjorov P. and Mladenov I., Cylindrical Equilibrium Shapes of Fluid Membranes, J. Phys. A: Math. & Theor. 41 (2008) 435201 (16pp); doi:10.1088/1751-8113/41/43/435201. 10.1088/1751-8113/41/43/435201
Search in Google Scholar

Articles in this issue

DOI: https://doi.org/10.2478/v10309-012-0042-3 | Journal eISSN: 1844-0835 | Journal ISSN: 1224-1784

Journal RSS Feed

Language: English

Page range: 79 - 88

Published on: May 17, 2013

Published by: Ovidius University of Constanta

In partnership with: Paradigm Publishing Services

Publication frequency: 3 issues per year

Keywords:

Delaunay surfaces,

membranes,

general shape equation

Related subjects:

Mathematics,

General mathematics

© 2013 Ivaïlo M. Mladenov, Mariana Ts. Hadzhilazova, published by Ovidius University of Constanta
This work is licensed under the Creative Commons License.

Volume 20 (2012): Issue 2 (June 2012)