Embeddedness of min-max CMC hypersurfaces in manifolds with positive Ricci curvature
Embeddedness of min-max CMC hypersurfaces in manifolds with positive Ricci curvature
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正里奇曲率流形中最小-最大 CMC 超曲面的嵌入
DOI：
10.1007/s00030-023-00910-7
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发表时间：
2024
期刊：
影响因子：
--
通讯作者：
Bellettini C
中科院分区：
文献类型：
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作者：
Bellettini C
我们证明，在具有正 Ricci 曲率的 3 维或更高维的紧致黎曼流形上，Bellettini 和 Wickramasekera 中的 Allen-Cahn 最小-最大方案（非齐次 Allen-Cahn 方程和规定平均曲率超曲面的存在，2020），规定函数被视为非零常数，产生恒定平均曲率（-CMC）的嵌入超曲面。更准确地说，我们证明由所述最小值-最大值产生的界面不包含偶重数最小超曲面，也不包含准嵌入点（这两种情况在 Bellettini 和 Wickramasekera，2020 年的结论中原则上都是可能的）。直接的几何推论是对于任何规定的非零常数，存在嵌入的闭合 CMC 超曲面（莫尔斯指数为 1）（在上述环境流形中），当环境维数为 8 或更高时，具有预期的奇异集。
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, produces an embedded hypersurface of constant mean curvature(-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-CMC hypersurfaces (with Morse index 1) for any prescribed non-zero constant, with the expected singular set when the ambient dimension is 8 or higher.