The Allen-Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet separation estimates for stable solutions (building on recent work of Wang-Wei) of the Allen-Cahn equation on a 3-manifold. Using these, we are able to show for generic metrics on a 3-manifold, minimal surfaces arising from Allen-Cahn solutions with bounded energy and bounded Morse index are two-sided and occur with multiplicity one and the expected Morse index. This confirms, in the Allen-Cahn setting, a strong form of the multiplicity one conjecture and the index lower bound conjecture of Marques-Neves in 3-dimensions regarding min-max constructions of minimal surfaces. Allen-Cahn min-max constructions were recently carried out by Guaraco and Gaspar-Guaraco. Our resolution of the multiplicity one and the index lower bound conjectures shows that these constructions can be applied to give a new proof of Yau's conjecture on infinitely many minimal surfaces in a 3-manifold with a generic metric (recently proven by Irie-Marques-Neves) with new geometric conclusions. Namely, we prove that a 3-manifold with a generic metric contains, for every $p$ = 1, 2, 3, ..., a two-sided embedded minimal surface with Morse index $p$ and area $p^{1/3}$, as conjectured by Marques-Neves.

中文翻译：

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最小曲面和 3 流形上的 Allen-Cahn 方程：指数、多重性和曲率估计

Allen-Cahn 方程是一个半线性偏微分方程，它通过奇异极限与极小超曲面理论密切相关。我们证明了 3 流形上 Allen-Cahn 方程的稳定解（建立在 Wang-Wei 的最近工作的基础上）的曲率估计和强片分离估计。使用这些，我们能够显示 3 流形上的通用度量，由具有有界能量和有界莫尔斯指数的 Allen-Cahn 解产生的最小曲面是两侧的，并且以多重性 1 和预期的莫尔斯指数出现。这证实了在 Allen-Cahn 设置中，关于最小曲面的最小-最大构造的 3 维中的多重性一猜想和指数下界猜想的强形式。最近由 Guaraco 和 Gaspar-Guaraco 进行了 Allen-Cahn 最小-最大构造。我们对重数一和指数下界猜想的解析表明，这些构造可用于在具有通用度量的 3 流形中的无限多个极小曲面上给出 Yau 猜想的新证明（最近由 Irie-Marques-Neves 证明） ) 与新的几何结论。也就是说，我们证明了具有通用度量的 3 流形包含，对于每个 $p$ = 1, 2, 3, ...，一个两侧嵌入的最小曲面，具有莫尔斯指数 $p$ 和面积 $p^{ 1/3}$，由 Marques-Neves 推测。