Regularity theory for $2$-dimensional almost minimal currents I: Lipschitz approximation

TL;DR

This paper develops a method to approximate 2-dimensional almost minimal currents with Lipschitz Q-valued functions, facilitating analysis of their singular sets in geometric measure theory.

Contribution

It introduces a Lipschitz approximation technique for integral currents with small excess, advancing the regularity theory for almost minimal currents.

Findings

Lipschitz Q-valued functions effectively approximate integral currents

The method aids in proving the discreteness of singular sets

Applicable to area minimizing, semicalibrated, and spherical cross sections

Abstract

We construct Lipschitz $Q$-valued functions which approximate carefully integral currents when their cylindrical excess is small and they are almost minimizing in a suitable sense. This result is used in two subsequent works to prove the discreteness of the singular set for the following three classes of $2$-dimensional integral currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of $3$-dimensional area minimizing cones.