Proceedings of the International Conference on Geometry, Integrability and Quantization

Functionals on Toroidal Surfaces

Metin Gürses

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Abstract

We show that the torus in ${\mathbb R}^3$ is a critical point of a sequence of functionals ${\mathcal F}_{n}$ ($n=1,2,3, \ldots$) defined over compact surfaces (closed membranes) in ${\mathbb R}^3$. When the Lagrange function ${\mathcal E}$ is a polynomial of degree $n$ of the mean curvature $H$ of the torus, the radii ($a,r$) of the torus are constrained to satisfy $\frac{a^2}{r^2}=\frac{n^2-n}{n^2-n-1},~~ n \ge 2$. A simple generalization of torus in ${\mathbb R}^3$ is a tube of radius $r$ along a curve ${\bf \alpha}$ which we call it toroidal surface (TS). We show that toroidal surfaces with non-circular curve ${\bf \alpha}$ do not provide minimal energy surfaces of the functionals ${\mathcal F}_{n}$ ($n=2,3$) on closed surfaces. We discuss possible applications of the functionals discussed in this work on cell membranes.

Article information

Source
Proceedings of the Seventeenth International Conference on Geometry, Integrability and Quantization, Ivaïlo M. Mladenov, Guowu Meng and Akira Yoshioka, eds. (Sofia: Avangard Prima, 2016), 270-283

Dates
First available in Project Euclid: 15 December 2015

Permanent link to this document
https://projecteuclid.org/ euclid.pgiq/1450194162

Digital Object Identifier
doi:10.7546/giq-17-2016-270-283

Mathematical Reviews number (MathSciNet)
MR3445435

Zentralblatt MATH identifier
1346.53012

Citation

Gürses, Metin. Functionals on Toroidal Surfaces. Proceedings of the Seventeenth International Conference on Geometry, Integrability and Quantization, 270--283, Avangard Prima, Sofia, Bulgaria, 2016. doi:10.7546/giq-17-2016-270-283. https://projecteuclid.org/euclid.pgiq/1450194162


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