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A Helfrich functional for compact surfaces in $\mathbb{C}P^{2}$

Part of: Global differential geometry Classical differential geometry

Published online by Cambridge University Press:  04 October 2023

Zhongwei Yao
Zhongwei Yao*
Affiliation:
School of Mathematics and Statistics, Fujian Normal University, Fuzhou, 350117, P. R. China School of Mathematics and Physics, Leshan Normal University, Leshan, Sichuan, 614004, P. R. China
*
Email: [email protected]
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Abstract

Let $f\;:\; M\rightarrow \mathbb{C}P^{2}$ be an isometric immersion of a compact surface in the complex projective plane $\mathbb{C}P^{2}$. In this paper, we consider the Helfrich-type functional $\mathcal{H}_{\lambda _{1},\lambda _{2}}(f)=\int _{M}(|H|^{2}+\lambda _{1}+\lambda _{2} C^{2})\textrm{d} M$, where $\lambda _{1}, \lambda _{2}\in \mathbb{R}$ with $\lambda _{1}\geqslant 0$, $H$ and $C$ are respectively the mean curvature vector and the Kähler function of $M$ in $\mathbb{C}P^{2}$. The critical surfaces of $\mathcal{H}_{\lambda _{1},\lambda _{2}}(f)$ are called Helfrich surfaces. We compute the first variation of $\mathcal{H}_{\lambda _{1},\lambda _{2}}(f)$ and classify the homogeneous Helfrich tori in $\mathbb{C}P^{2}$. Moreover, we study the Helfrich energy of the homogeneous tori and show the lower bound of the Helfrich energy for such tori.

Keywords

Helfrich functionalKähler anglehomogeneous tori

MSC classification

Primary: 53A05: Surfaces in Euclidean space
Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.)
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 66 , Issue 1 , January 2024 , pp. 36 - 50
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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