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Journal Article
A Helfrich functional for compact surfaces in
2024
Overview
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.
Publisher
Cambridge University Press
Subject
