AI 中文总结
研究闭黎曼流形上等变艾伦 - 卡恩解的正则性理论，利用修正方程和相关理论，证明特定条件下解收敛到极小超曲面，还扩展了变分构造，得到等变艾伦 - 卡恩p - 宽度收敛结果，为相关流形存在极小超曲面提供依据。
AI 中文摘要
我们为闭黎曼流形上具有等距作用的李群的等变艾伦 - 卡恩解发展了一种正则性理论。当作用的上同调维数在3到7之间时，我们证明了一列具有一致有界能量和等变指标的等变艾伦 - 卡恩解收敛到具有最优正则性的嵌入极小超曲面，即奇异集至少为余维数7且位于所有非主轨道的并集中。当上同调维数为2且作用没有例外轨道时，我们得到相同结果，但极小超曲面可能是浸入的。因此，任何具有上同调维数为2的李群作用且无例外轨道的闭黎曼流形都允许存在具有最优正则性的极小超曲面。一个关键工具是基于王 - 魏的工作的乔多什 - 曼图利迪斯正则性理论。然而，我们将他们的论证应用于带有漂移拉普拉斯算子的修正艾伦 - 卡恩方程。我们还表明，当极限极小超曲面光滑时，对于其有适当的指标界。我们还通过定义等变山路不变量以及等变艾伦 - 卡恩p - 宽度，扩展了瓜拉科和加斯帕尔 - 瓜拉科的艾伦 - 卡恩方程解的变分构造。这基于格罗莫夫的工作，是阿尔姆格伦 - 皮茨设定下艾伦 - 卡恩与王的等变体积谱的平行内容。我们证明，当ε趋于0时，等变艾伦 - 卡恩p - 宽度收敛到王定义的等变p - 宽度。
英文摘要
We develop a regularity theory for equivariant Allen--Cahn solutions on closed Riemannian manifolds with a Lie group acting isometrically. When the cohomogeneity of the action is between $3$ and $7$, we show that a sequence of equivariant Allen--Cahn solutions with uniformly bounded energy and equivariant index converge to embedded minimal hypersurfaces with optimal regularity, meaning that the singular set is at least codimension $7$ and lies in the union of all non-principal orbits. When the cohomogeneity is $2$ and the action has no exceptional orbits, we show the same result but the minimal hypersurfaces may be immersed. As a result, any closed Riemmanian manifold with cohomogeneity $2$ Lie group action and no exceptional orbits admits a minimal hypersurface with optimal regularity. A key tool is the regularity theory of Chodosh--Mantoulidis, building on the work of Wang--Wei. However, we adapt their arguments to a modified Allen--Cahn equation with a drift Laplacian. We also show that appropriate index bounds hold for the limiting minimal hypersurface when it is smooth. We also extend the variational constructions of solutions of the Allen--Cahn equation of Guaraco and Gaspar--Guaraco by defining an equivariant mountain pass invariant, as well as the equivariant Allen--Cahn $p$-widths. This builds on the work of Gromov and is the Allen--Cahn parallel to Wang's equivariant volume spectrum in the Almgren-Pitts setting. We show that in the limit as $ε$ tends to $0$, the equivariant Allen--Cahn $p$-widths converge to the equivariant $p$-widths, as defined by Wang.
Comments44 pages, 2 figures, comments welcome! V2: fixed notation in theorem 1.1, modified remark 6.3, added acknowledgements. No major changes
