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| Format: | Preprint |
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2020
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| Online Access: | https://arxiv.org/abs/2004.05120 |
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| _version_ | 1866913397116764160 |
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| author | Dey, Akashdeep |
| author_facet | Dey, Akashdeep |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2004_05120 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A comparison of the Almgren-Pitts and the Allen-Cahn min-max theory Dey, Akashdeep Differential Geometry 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. |
| title | A comparison of the Almgren-Pitts and the Allen-Cahn min-max theory |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2004.05120 |