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

TL;DR

This paper provides an extensive, accessible overview of the development and current state of the regularity theory for generalized minimal surfaces within geometric measure theory, highlighting key milestones and ongoing challenges.

Contribution

It offers a comprehensive, nontechnical account of the evolution, recent advances, and unresolved issues in the regularity theory for minimal surfaces.

Findings

Historical overview of regularity theory development

Recent advancements in the field

Identification of remaining challenges

Abstract

The aim of this article is to give a rather extensive, and yet nontechnical, account of the birth of the regularity theory for generalized minimal surfaces, of its various ramifications along the decades, of the most recent developments, and of some of the remaining challenges.