Minimal surfaces and the Allen–Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates
Minimal surfaces and the Allen–Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates
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DOI：
10.4007/annals.2020.191.1.4
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发表时间：
2018-03
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影响因子：
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通讯作者：
Otis Chodosh;Christos Mantoulidis
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作者：
Otis Chodosh;Christos Mantoulidis
Allen-Cahn 方程是一个半线性偏微分方程，它通过奇异极限与最小超曲面理论紧密相连。我们证明了 3 流形上 Allen-Cahn 方程稳定解的曲率估计和强片分离估计（基于 Wang-Wei 最近的工作）。使用这些，我们能够证明 3 流形上的通用度量，由具有有限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，并且以重数 1 和预期莫尔斯指数出现。这在 Allen-Cahn 设置中证实了关于最小曲面的最小-最大构造的 3 维重数一猜想和 Marques-Neves 指数下界猜想的强形式。 Guaraco 和 Gaspar-Guaraco 最近进行了 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，这些构造可以应用于用通用度量（最近由 Irie-Marques-Neves 证明）和新的几何结论，对 3 流形中的无限多个最小曲面上的丘猜想给出新的证明。也就是说，我们证明，对于每个 $p$ = 1, 2, 3, ...，具有通用度量的 3 流形包含一个具有莫尔斯索引 $p$ 和面积 $p^{1/3}$ 的双边嵌入最小曲面，正如 Marques-Neves 所推测的那样。
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.