1 results
A Helfrich functional for compact surfaces in
$\mathbb{C}P^{2}$
- Part of
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- Journal:
- Glasgow Mathematical Journal / Volume 66 / Issue 1 / January 2024
- Published online by Cambridge University Press:
- 04 October 2023, pp. 36-50
- Print publication:
- January 2024
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- Article
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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.
