{"title":"各向异性最小-最大理论:各向异性极小面和CMC面的存在性","authors":"Guido De Philippis, Antonio De Rosa","doi":"10.1002/cpa.22189","DOIUrl":null,"url":null,"abstract":"<p>We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth <span></span><math>\n <semantics>\n <mrow>\n <mn>3</mn>\n </mrow>\n <annotation>$\\hskip.001pt 3$</annotation>\n </semantics></math>–dimensional Riemannian manifolds. The constructed min-max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard in dimension <span></span><math>\n <semantics>\n <mrow>\n <mn>3</mn>\n </mrow>\n <annotation>$\\hskip.001pt 3$</annotation>\n </semantics></math>.</p>","PeriodicalId":3,"journal":{"name":"ACS Applied Electronic Materials","volume":null,"pages":null},"PeriodicalIF":4.3000,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22189","citationCount":"0","resultStr":"{\"title\":\"The anisotropic min-max theory: Existence of anisotropic minimal and CMC surfaces\",\"authors\":\"Guido De Philippis, Antonio De Rosa\",\"doi\":\"10.1002/cpa.22189\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth <span></span><math>\\n <semantics>\\n <mrow>\\n <mn>3</mn>\\n </mrow>\\n <annotation>$\\\\hskip.001pt 3$</annotation>\\n </semantics></math>–dimensional Riemannian manifolds. The constructed min-max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard in dimension <span></span><math>\\n <semantics>\\n <mrow>\\n <mn>3</mn>\\n </mrow>\\n <annotation>$\\\\hskip.001pt 3$</annotation>\\n </semantics></math>.</p>\",\"PeriodicalId\":3,\"journal\":{\"name\":\"ACS Applied Electronic Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.3000,\"publicationDate\":\"2023-12-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22189\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Electronic Materials\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/cpa.22189\",\"RegionNum\":3,\"RegionCategory\":\"材料科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"ENGINEERING, ELECTRICAL & ELECTRONIC\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Electronic Materials","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/cpa.22189","RegionNum":3,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"ENGINEERING, ELECTRICAL & ELECTRONIC","Score":null,"Total":0}
The anisotropic min-max theory: Existence of anisotropic minimal and CMC surfaces
We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth –dimensional Riemannian manifolds. The constructed min-max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard in dimension .
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