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.

We consider a Canham − Helfrich − type variational problem defined over closed surfacesenclosing a fixed volume and having fixed surface area. The problem models the shape ofmultiphase biomembranes. It consists of minimizing the sum of the Canham − Helfrichenergy, in which the bending rigidities and spontaneous curvatures are nowphase-dependent, and a line tension penalization for the phase interfaces. By restrictingattention to axisymmetric surfaces and phase distributions, we extend our previous resultsfor a single phase [R. Choksi and M. Veneroni, Calc. Var. Partial Differ. Equ.(2012). DOI:10.1007/s00526-012-0553-9] and prove existence of a globalminimizer.