论文标题
在具有阳性RICCI曲率的歧管中，Multerity-1 Minmax最小超曲面
Multiplicity-1 minmax minimal hypersurfaces in manifolds with positive Ricci curvature
论文作者
论文摘要
我们通过Allen-Cahn Energy解决了单参数Minmax的结构，该结构最近导致了新的证据证明在任意紧凑的Riemannian歧管$ n^{n+1} $中，使用$ n \ geq 2 $（请参阅Guaraco的2018年工作）。我们获得以下多重性 - $ 1 $结果：如果$ n $的RICCI曲率为正，则Minmax Allen-cahn Solutions集中在多重性-1 $ hypersurface周围 - $ 1 $ hypersurface，可能具有一组奇异的尺寸$ \ leq n-7 $。对于$ n \ geq 3 $（对于$ n = 2 $），此结果是新的，这也是Chodosh-Mantoulidis最近的工作所暗示的）。这里开发的论点是风味的几何，并直接利用了解决方案的Minmax表征。即时的推论是，每个紧凑的Riemannian歧管$ n^{n+1} $，带有$ n \ geq 2 $，正ricci曲率接收一个两面封闭的最小脱落的脱落面，可能最多有$ n-7 $的单数尺寸。这种存在的结果也来自多样性 - $ 1 $结果在Almgren-pitts框架内开发的结果，请参阅Ketover-Marques-neves，Zhou，Zhou，Marques-neves，Ramirez-Luna的作品。
We address the one-parameter minmax construction, via Allen--Cahn energy, that has recently lead to a new proof of the existence of a closed minimal hypersurface in an arbitrary compact Riemannian manifold $N^{n+1}$ with $n\geq 2$ (see Guaraco's 2018 work). We obtain the following multiplicity-$1$ result: if the Ricci curvature of $N$ is positive then the minmax Allen--Cahn solutions concentrate around a multiplicity-$1$ hypersurface, that may have a singular set of dimension $\leq n-7$. This result is new for $n\geq 3$ (for $n=2$ it is also implied by the recent work by Chodosh--Mantoulidis). The argument developed here is geometric in flavour and exploits directly the minmax characterization of the solutions. An immediate corollary is that every compact Riemannian manifold $N^{n+1}$ with $n\geq 2$ and positive Ricci curvature admits a two-sided closed minimal hypersurface, possibly with a singular set of dimension at most $n-7$. This existence result also follows from multiplicity-$1$ results developed within the Almgren--Pitts framework, see works by Ketover-Marques-Neves, Zhou, Marques-Neves, Ramirez-Luna.
