2014/01/29 by Spencer T. Becker-Kahn, Becker-Kahn, Spencer T.
Engineering · Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #Structural Analysis and Optimization
paper · pdf · doi:10.48550/arxiv.1401.7660
openalex publication_date 2014/01/29 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
We prove some epsilon regularity results for n-dimensional minimal two-valued\nLipschitz graphs. The main theorems imply uniqueness of tangent cones and\nregularity of the singular set in a neighbourhood of any point at which at\nleast one tangent cone is equal to a pair of transversely intersecting\nmultiplicity one n-dimensional planes, and in a neighbourhood of any point at\nwhich at which at least one tangent cone is equal to a union of four distinct\nmultiplicity one n-dimensional half-planes that meet along an (n-1) -\ndimensional axis. The key ingredient is a new Excess Improvement Lemma obtained\nvia a blow-up method (inspired by the work of L. Simon on the singularities of\n`multiplicity one' classes of minimal submanifolds) and which can be iterated\nunconditionally. We also show that any tangent cone to an n-dimensional minimal\ntwo-valued Lipschitz graph that is translation invariant along an (n-1) or\n(n-2)- dimensional subspace is indeed a cone of one of the two aforementioned\nforms, which yields a global decomposition result for the singular set\n