论文标题
Almgren-Pitts和Allen-Cahn Min-Max理论的比较
A comparison of the Almgren-Pitts and the Allen-Cahn min-max theory
论文作者
论文摘要
Allen-Cahn方程的最低理论是由Guaraco和Gaspar-Guaraco开发的。他们表明，艾伦-CAHN宽度大于或等于Almgren-Pitts宽度。在本文中，我们将证明反向不等式也存在，即Allen-Cahn宽度小于或等于Almgren-Pitts宽度。因此，Almgren-Pitts的宽度和Allen-Cahn宽度一致。我们还将表明，从Allen-Cahn Min-Max理论获得的所有封闭的最小超曲面（具有最佳的规律性）也由Almgren-Pitts Min-Max理论产生。结果，我们将指出，Marques-neves和Li证明的Almgren-Pitts环境中的指数上限也可以从Allen-Cahn环境中的指数上限获得，并由Gaspar和Hiesmayr证明。
Min-max theory for the Allen-Cahn equation was developed by Guaraco and Gaspar-Guaraco. They showed that the Allen-Cahn widths are greater than or equal to the Almgren-Pitts widths. In this article we will prove that the reverse inequalities also hold i.e. the Allen-Cahn widths are less than or equal to the Almgren-Pitts widths. Hence, the Almgren-Pitts widths and the Allen-Cahn widths coincide. We will also show that all the closed minimal hypersurfaces (with optimal regularity) which are obtained from the Allen-Cahn min-max theory are also produced by the Almgren-Pitts min-max theory. As a consequence, we will point out that the index upper bound in the Almgren-Pitts setting, proved by Marques-Neves and Li, can also be obtained from the index upper bound in the Allen-Cahn setting, proved by Gaspar and Hiesmayr.
