{"title":"Embeddedness of min–max CMC hypersurfaces in manifolds with positive Ricci curvature","authors":"Costante Bellettini, Myles Workman","doi":"10.1007/s00030-023-00910-7","DOIUrl":null,"url":null,"abstract":"<p>We prove that on a compact Riemannian manifold of dimension 3 or higher, with positive Ricci curvature, the Allen–Cahn min–max scheme in Bellettini and Wickramasekera (The Inhomogeneous Allen–Cahn Equation and the Existence of Prescribed-Mean-Curvature Hypersurfaces, 2020), with prescribing function taken to be a non-zero constant <span>\\(\\lambda \\)</span>, produces an embedded hypersurface of constant mean curvature <span>\\(\\lambda \\)</span> (<span>\\(\\lambda \\)</span>-CMC). More precisely, we prove that the interface arising from said min–max contains no even-multiplicity minimal hypersurface and no quasi-embedded points (both of these occurrences are in principle possible in the conclusions of Bellettini and Wickramasekera, 2020). The immediate geometric corollary is the existence (in ambient manifolds as above) of embedded, closed <span>\\(\\lambda \\)</span>-CMC hypersurfaces (with Morse index 1) for any prescribed non-zero constant <span>\\(\\lambda \\)</span>, with the expected singular set when the ambient dimension is 8 or higher.</p>","PeriodicalId":501665,"journal":{"name":"Nonlinear Differential Equations and Applications (NoDEA)","volume":"23 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinear Differential Equations and Applications (NoDEA)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00030-023-00910-7","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We prove that on a compact Riemannian manifold of dimension 3 or higher, with positive Ricci curvature, the Allen–Cahn min–max scheme in Bellettini and Wickramasekera (The Inhomogeneous Allen–Cahn Equation and the Existence of Prescribed-Mean-Curvature Hypersurfaces, 2020), with prescribing function taken to be a non-zero constant \(\lambda \), produces an embedded hypersurface of constant mean curvature \(\lambda \) (\(\lambda \)-CMC). More precisely, we prove that the interface arising from said min–max contains no even-multiplicity minimal hypersurface and no quasi-embedded points (both of these occurrences are in principle possible in the conclusions of Bellettini and Wickramasekera, 2020). The immediate geometric corollary is the existence (in ambient manifolds as above) of embedded, closed \(\lambda \)-CMC hypersurfaces (with Morse index 1) for any prescribed non-zero constant \(\lambda \), with the expected singular set when the ambient dimension is 8 or higher.
我们证明,在维度为 3 或更高且具有正里奇曲率的紧凑黎曼流形上,Bellettini 和 Wickramasekera (The Inhomogeneous Allen-Cahn Equation and the Existence of Prescribed-Mean-Curvature Hypersurfaces.)的 Allen-Cahn 最小-最大方案会产生一个内嵌的具有恒定平均曲率的超曲面、2020)中,规定函数被认为是一个非零常数(\(\lambda \)),产生了一个内嵌的恒定平均曲率超曲面(\(\lambda \)-CMC)。更准确地说,我们证明了由上述最小-最大产生的界面不包含偶数多重性最小超曲面和准嵌入点(这两种情况在贝莱蒂尼和维克拉马塞克拉的结论中原则上都是可能的,2020)。紧接着的几何推论是,对于任何规定的非零常数\(\lambda \),当环境维度为8或更高时,存在内嵌的、封闭的\(\lambda \)-CMC超曲面(莫尔斯指数为1),并具有预期奇异集(在环境流形中如上所述)。