A comparison of the Almgren–Pitts and the Allen–Cahn min–max theory

Abstract

Min–max theory for the Allen–Cahn equation was developed by Guaraco (J Differ Geom 108:91–133, 2018) and Gaspar–Guaraco (Calc Var Partial Differ Equ 57:101, 42, 2018). 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 (Camb J Math 4(4):463–511, 2016) and Li (An improved Morse index bound of min–max minimal hypersurfaces, 2020. arXiv:2007.14506 [math.DG]), can also be obtained from the index upper bound in the Allen–Cahn setting, proved by Gaspar (J Geom Anal 30:69-85, 2020) and Hiesmayr (Commun Partial Differ Equ 43(11):1541–1565, 2018).