## Proceedings of the International Conference on Geometry, Integrability and Quantization

- Geom. Integrability & Quantization
- 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

### Functionals on Toroidal Surfaces

#### 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

**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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