Regularity theory for $2$-dimensional almost minimal currents III: blowup

TL;DR

This paper investigates the asymptotic behavior of 2-dimensional almost minimal currents at singular points, contributing to understanding the structure and discreteness of their singular sets in various geometric contexts.

Contribution

It advances regularity theory by analyzing blowup limits of almost minimal currents, establishing discreteness of singularities for specific classes of 2D currents.

Findings

Discreteness of the singular set for area minimizing currents

Asymptotic analysis of blowup limits at singular points

Characterization of singularities in semicalibrated and spherical cross sections

Abstract

We analyze the asymptotic behavior of a $2$-dimensional integral current which is almost minimizing in a suitable sense at a singular point. Our analysis is the second half of an argument which shows the discreteness of the singular set for the following three classes of $2$-dimensional currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of $3$-dimensional area minimizing cones.