论文标题
通过allen-cahn的多种多数1 minmax最小超曲面的通用存在
Generic existence of multiplicity-1 minmax minimal hypersurfaces via Allen--Cahn
论文作者
论文摘要
在瓜拉科（Guaraco）的2018年作品中，有一个新的证明，证明了紧凑型riemannian歧管$ n^{n+1} $的封闭最小的超浮雕，并带有$ n \ geq 2 $。这是通过采用艾伦（Allen）近似方案和艾伦（Allen）能量的单参数minmax（依靠哈钦森（Hutchinson），tonegawa，wickramasekera的作品来实现的，因为艾伦·卡纳（Allen-Cahn）参数倾向于$ 0 $）。获得的最小超出表面可以先验携带局部恒定整数多重性。在这里，我们考虑了一种Minmax结构，它是对上述作品中的修改，它可以在选择山脉通行证之间的山谷点进行初步自由，然后对所述选择进行优化。我们证明，当$ 2 \ leq n \ leq 6 $且度量标准时，MinMax会导致（平滑的封闭）最小超曲面，其多重性$ 1 $。 （当$ n = 2 $当Chodosh的结论中也得到了结论。通过Almgren--pitts理论）。
In Guaraco's 2018 work a new proof was given of the existence of a closed minimal hypersurface in a compact Riemannian manifold $N^{n+1}$ with $n\geq 2$. This was achieved by employing an Allen--Cahn approximation scheme and a one-parameter minmax for the Allen--Cahn energy (relying on works by Hutchinson, Tonegawa, Wickramasekera to pass to the limit as the Allen-Cahn parameter tends to $0$). The minimal hypersurface obtained may a priori carry a locally constant integer multiplicity. Here we consider a minmax construction that is a modification of the one in the aforementioned work, by allowing an initial freedom on the choice of the valley points between which the mountain pass construction is carried out, and then optimising over said choice. We prove that, when $2\leq n\leq 6$ and the metric is bumpy, this minmax leads to a (smooth closed) minimal hypersurface with multiplicity $1$. (When $n=2$ this conclusion also follows from Chodosh--Mantoulidis's recent work.) As immediate corollary we obtain that every compact Riemannian manifold of dimension $n+1$, $2\leq n\leq 6$, endowed with a bumpy metric, admits a two-sided smooth closed minimal hypersurface (this existence conclusion also follows from Zhou's recent result for minmax constructions via Almgren--Pitts theory).
