具有正里奇曲率的流形中的重数1极大极小超曲面
Multiplicity-1 minmax minimal hypersurfaces in manifolds with positive Ricci curvature
摘要：
我们通过Allen-Cahn能量解决了单参数极小极大构造，最近导致了一个新的证明，证明了在任意紧致黎曼流形$N^{n+1}$中存在一个封闭的极小超曲面，其中$n\geq 2$（参见Guaraco 2018年的工作）。我们得到了如下的重数$1$结果：如果$N$的里奇曲率为正，则极小极大Allen-Cahn解集中在重数$1$超曲面上，该超曲面可能具有一个奇异的$\leq n-7$维集.这一结果对于$n\geq 3$是新的（对于$n=2$，Chodosh-Mantoulidis最近的工作也暗示了这一点）。在这里开发的参数是几何的味道和直接利用极小极大的解决方案的特点。一个直接的推论是：每个具有$n\geq 2$和正里奇曲率的紧致黎曼流形$N^{n+1}$都有一个双边闭极小超曲面，可能有一个奇异维数集至多$n-7$ .这个存在的结果也遵循多重性- $1$的结果内发展的Almgren-皮茨框架，见作品由Ketover-Marques-Neves，周，Marques-Neves，拉米雷斯-卢纳。
英文摘要：
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.
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