Multiple valued functions and integral currents

TL;DR

This paper explores the mathematical relationship between Almgren's multiple valued functions and integral currents, providing new proofs and explicit formulas that advance understanding in geometric measure theory.

Contribution

It offers a simplified proof that Lipschitz multiple valued maps define integer rectifiable currents and derives explicit formulas for their boundary, mass, and variations.

Findings

Lipschitz multiple valued maps naturally define integer rectifiable currents

Explicit formulas for boundary, mass, and first variations are established

A reparametrization property for multiple valued functions is derived

Abstract

We prove several results on Almgren's multiple valued functions and their links to integral currents. In particular, we give a simple proof of the fact that a Lipschitz multiple valued map naturally defines an integer rectifiable current; we derive explicit formulae for the boundary, the mass and the first variations along certain specific vector-fields; and exploit this connection to derive a delicate reparametrization property for multiple valued functions. These results play a crucial role in our new proof of the partial regularity of area minimizing currents.