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

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.