Sliding stability and uniqueness for the set Y × Y

Abstract

This article is dedicated to discuss the sliding stability and the uniqueness property for the 2-dimensional minimal cone Y×Y in R4. This problem is motivated by the classification of singularities for Almgren minimal sets, a model for Plateau's problem in the setting of sets. Minimal cones are blow up limits of Almgren minimal sets, thus the list of all minimal cones gives all possible types of singularities that can occur for minimal sets.As proved in [16], when several 2-dimensional Almgren (resp. topological) minimal cones are Almgren (resp. topological) sliding stable, and Almgren (resp. topological) unique, the almost orthogonal union of them stays minimal. Hence if several minimal cones admit sliding stability and uniqueness properties, then we can use their almost orthogonal unions to generate new families of minimal cones. One then naturally ask which minimal cones admit these two properties.This list of known 2-dimensional minimal cones in arbitrary ambient dimension is not long, and the stability and uniqueness properties for all the known 2-dimensional minimal cones, except for Y×Y, have already been established in the previous works [18,17]. Among all the known 2-dimensional minimal cones, Y×Y is the only one whose stability and uniqueness properties were left unsolved. This is due to two main reasons: 1) Y×Y is the only known minimal cone which is essentially of codimension larger than 1—that is, we cannot decompose it into transversal unions of minimal cones of codimension 1. 2) Y×Y lives in R4, where we know very little about which types of singularities can occur in a minimal set. This makes it difficult to control and estimate the measures of all possible competitors. Due to the above two issues, new ideas are required here for solving the problem.We give affirmative answers to this problem for the stability and uniqueness properties for Y×Y in this paper: we prove that the set Y×Y is both Almgren sliding stable, and Almgren unique; for the topological case, we prove its topological sliding stability and topological uniqueness for the coefficient group Z2. This result, along with the results in [16,18,17], allows us to use all the known 2-dimensional minimal cones to generate new 2-dimensional minimal cones by taking almost orthogonal unions.