Uniqueness of Tangent Cones for Two‐Dimensional Almost‐Minimizing Currents

Abstract

We consider two‐dimensional integer rectifiable currents that are almost area minimizing and show that their tangent cones are everywhere unique. Our argument unifies a few uniqueness theorems of the same flavor, which are all obtained by a suitable modification of White's original theorem for area‐minimizing currents in the euclidean space. This note is also the first step in a regularity program for semicalibrated two‐dimensional currents and spherical cross sections of three‐dimensional area‐minimizing cones.© 2017 Wiley Periodicals, Inc.

The regularity theory for the area functional (in geometric measure theory)

“…Based on the works [47,48,49,50,51] the author, Spadaro, and Luca Spolaor gave a complete independent proof of the existence of a branched center manifold in [53]. We also developed a suitable more general counterpart of Chang's theory in the papers [52,54,55], proving in particular the same regularity result for spherical cross-sections of area-minimizing 3-dimensional cones and for semicalibrated 2-dimensional currents (previous theorems in [12,13] proved some cases of particular interest, based on the works of Rivière and Tian, see [99,100,101]). Almgren's dimension bound in Theorem 10.1 has as well been extended to semicalibrated currents by Spolaor in [113].…”

“…Simone Steinbrüchel in [43,44], building in part upon the theory developed in [34] and the paper [73]. Theorem 14.3.…”

The size of the singular set of area-minimizing currents

“…Building upon [26,27,30,28,29], in forthcoming joint papers with Emanuele Spadaro and Luca Spolaor we will give the first proof of the existence of a "branched center manifold" and extend Chang's theorem to a large class of objects which are almost minimizing in a suitable sense, cf. [34,31,32,33]. That proof (and Chang's theorem) will however not be discussed in this survey.…”

Boundary Regularity of Mass-Minimizing Integral Currents and a Question of Almgren

“…[11,13,12,14,15]) the first author and Emanuele Spadaro have revisited Almgren's theory introducing several new ideas which simplify his proof considerably. Furthermore, the first author together with Spadaro and Spolaor, in [19,16,18,17] applied these sets of ideas to establish a complete proof of Chang's interior regularity results for 2 dimensional mass-minimizing currents [7], showing that in this case interior singular points are isolated.…”