Regularity of area minimizing currents III: blow-up

TL;DR

This paper completes a series providing a new, concise proof of Almgren's partial regularity for area minimizing currents, using blow-up analysis to relate their regularity to Dir-minimizing multiple valued functions.

Contribution

It introduces a simplified proof of Almgren's partial regularity theorem through blow-up analysis connecting currents to Dir-minimizing functions.

Findings

Established regularity of area minimizing currents via blow-up techniques.

Linked the regularity problem to Dir-minimizing multiple valued functions.

Provided a shorter, more accessible proof of a classical regularity result.

Abstract

This is the last of a series of three papers in which we give a new, shorter proof of a slightly improved version of Almgren's partial regularity of area minimizing currents in Riemannian manifolds. Here we perform a blow-up analysis deducing the regularity of area minimizing currents from that of Dir-minimizing multiple valued functions.