论文标题
Min-Max CMC Hypersurfaces的嵌入在具有阳性RICCI曲率的歧管中
Embeddedness of Min-Max CMC Hypersurfaces in Manifolds with Positive Ricci Curvature
论文作者
论文摘要
我们证明，在紧凑的riemannian歧管上，尺寸为$ 3 $ $ 3 $，带有正曲率曲率，allen-cahn-min-max方案（由第一作者实施，2020年N. wickramasekera在2020年），处方功能是非零常数的$λ$，可产生一个不变的hypersed $ culvercy $ curvate $ curvate $ curvate $（）更准确地说，我们证明，由上述最小值产生的界面不包含偶数性最小的高度表面，也不包含Quasi Embedded点（在第一作者和N. Wickramasekera的上述作品的结论中，这两种情况都是可能的。直接的几何推论是嵌入式，封闭的$λ$ -CMC Hypersurfaces（带有MORSE指数$ 1 $）的嵌入式，封闭的$λ$ -CMC Hypersurfaces的存在（在环境流形中），对于任何规定的非零常数$λ$），当周围尺寸为$ 8 $ 8 $ 8 $ 8 $ 8 $ 8 $ 8 $ 8 $λ$。
We prove that on a compact Riemannian manifold of dimension $3$ or higher, with positive Ricci curvature, the Allen--Cahn min-max scheme (implemented by the first author and N. Wickramasekera in 2020), with prescribing function taken to be a non-zero constant $λ$, produces an embedded hypersurface of constant mean curvature $λ$ ($λ$-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 the aforementioned work by the first author and N. Wickramasekera). The immediate geometric corollary is the existence (in ambient manifolds as above) of embedded, closed $λ$-CMC hypersurfaces (with Morse index $1$) for any prescribed non-zero constant $λ$, with the expected singular set when the ambient dimension is $8$ or higher.
