{"title":"A Helfrich functional for compact surfaces in","authors":"Zhongwei Yao","doi":"10.1017/s0017089523000320","DOIUrl":null,"url":null,"abstract":"<p>Let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231003124134699-0703:S0017089523000320:S0017089523000320_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$f\\;:\\; M\\rightarrow \\mathbb{C}P^{2}$</span></span></img></span></span> be an isometric immersion of a compact surface in the complex projective plane <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231003124134699-0703:S0017089523000320:S0017089523000320_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathbb{C}P^{2}$</span></span></img></span></span>. In this paper, we consider the Helfrich-type functional <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231003124134699-0703:S0017089523000320:S0017089523000320_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathcal{H}_{\\lambda _{1},\\lambda _{2}}(f)=\\int _{M}(|H|^{2}+\\lambda _{1}+\\lambda _{2} C^{2})\\textrm{d} M$</span></span></img></span></span>, where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231003124134699-0703:S0017089523000320:S0017089523000320_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$\\lambda _{1}, \\lambda _{2}\\in \\mathbb{R}$</span></span></img></span></span> with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231003124134699-0703:S0017089523000320:S0017089523000320_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$\\lambda _{1}\\geqslant 0$</span></span></img></span></span>, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231003124134699-0703:S0017089523000320:S0017089523000320_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$H$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231003124134699-0703:S0017089523000320:S0017089523000320_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$C$</span></span></img></span></span> are respectively the mean curvature vector and the Kähler function of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231003124134699-0703:S0017089523000320:S0017089523000320_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$M$</span></span></img></span></span> in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231003124134699-0703:S0017089523000320:S0017089523000320_inline10.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathbb{C}P^{2}$</span></span></img></span></span>. The critical surfaces of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231003124134699-0703:S0017089523000320:S0017089523000320_inline11.png\"/><span data-mathjax-type=\"texmath\"><span>$\\mathcal{H}_{\\lambda _{1},\\lambda _{2}}(f)$</span></span></span></span> are called Helfrich surfaces. We compute the first variation of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231003124134699-0703:S0017089523000320:S0017089523000320_inline12.png\"/><span data-mathjax-type=\"texmath\"><span>$\\mathcal{H}_{\\lambda _{1},\\lambda _{2}}(f)$</span></span></span></span> and classify the homogeneous Helfrich tori in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231003124134699-0703:S0017089523000320:S0017089523000320_inline13.png\"/><span data-mathjax-type=\"texmath\"><span>$\\mathbb{C}P^{2}$</span></span></span></span>. Moreover, we study the Helfrich energy of the homogeneous tori and show the lower bound of the Helfrich energy for such tori.</p>","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"88 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Glasgow Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/s0017089523000320","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
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.
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