We address the one-parameter minmax construction for the 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 with (Guaraco's work, relying on works by Hutchinson, Tonegawa, and Wickramasekera when sending the Allen–Cahn parameter to 0). We obtain the following result: if the Ricci curvature of N is positive then the minmax Allen–Cahn solutions concentrate around a multiplicity-1 minimal hypersurface (possibly having a singular set of dimension ). This multiplicity result is new for (for it is also implied by the recent work by Chodosh–Mantoulidis). We exploit directly the minmax characterization of the solutions and the analytic simplicity of semilinear (elliptic and parabolic) theory in . While geometric in flavour, our argument takes advantage of the flexibility afforded by the analytic Allen–Cahn framework, where hypersurfaces are replaced by diffused interfaces; more precisely, they are replaced by sufficiently regular functions (from N to ), whose weighted level sets give rise to diffused interfaces. We capitalise on the fact that (unlike a hypersurface) a function can be deformed both in the domain N (deforming the level sets) and in the target (varying the values). We induce different geometric effects on the diffused interface by using these two types of deformations; this enables us to implement in a continuous way certain operations, whose analogues on a hypersurface would be discontinuous. An immediate corollary of the multiplicity-1 conclusion is that every compact Riemannian manifold with and positive Ricci curvature admits a two-sided closed minimal hypersurface, possibly with a singular set of dimension at most . (This geometric corollary also follows from results obtained by different ideas in an Almgren–Pitts minmax framework.)

中文翻译：

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具有正 Ricci 曲率的流形中的 Multiplicity-1 minmax 最小超曲面

我们解决了艾伦-卡恩能量的单参数最小最大构造，该构造最近导致了任意紧致黎曼流形中存在闭合最小超曲面的新证明和（Guaraco 的工作，在将 Allen–Cahn 参数发送为 0 时，依赖于 Hutchinson、Tonekawa 和 Wickramasekera 的工作）。我们得到以下结果：如果N的 Ricci 曲率是正数，则最小最大 Allen–Cahn 解集中在重数为 1的最小超曲面周围（可能具有奇异的维数集））。这个多重结果是新的（为了Chodosh-Mantoulidis 最近的工作也暗示了这一点）。我们直接利用解的最小最大特征和半线性（椭圆形和抛物线）理论的分析简单性。虽然具有几何风格，但我们的论证利用了分析艾伦-卡恩框架提供的灵活性，其中超曲面被扩散界面所取代；更准确地说，它们被足够规则的函数取代（从N到），其加权水平集产生扩散界面。我们利用这样一个事实：（与超曲面不同）函数可以在域N（使水平集变形）和目标中变形（改变值）。我们通过使用这两种类型的变形在扩散界面上产生不同的几何效应；这使我们能够以连续的方式实施某些操作，而这些操作在超曲面上的类似操作将是不连续的。多重性 1 结论的直接推论是每个紧黎曼流形和正里奇曲率允许两侧闭合最小超曲面，最多可能具有一组奇异维数。（这个几何推论也是从 Almgren-Pitts 最小最大框架中的不同想法获得的结果得出的。）