# Multiple valued sections of vector bundles: the reparametrization theorem for $Q$-valued functions revisited

## TL;DR

This paper extends the concept of $Q$-valued functions to sections of vector bundles over Riemannian manifolds, providing new geometric insights and a simplified proof of a key reparametrization theorem crucial for regularity results in geometric measure theory.

## Contribution

It introduces a generalized notion of multiple valued sections of vector bundles and offers an elementary geometric proof of a fundamental reparametrization theorem.

## Key findings

- Generalization of $Q$-valued functions to vector bundle sections
- Elementary geometric proof of the reparametrization theorem
- Enhanced understanding of regularity in higher codimension currents

## Abstract

We analyze a notion of multiple valued sections of a vector bundle over an abstract smooth Riemannian manifold, which was suggested by W. Allard in the unpublished note "Some useful techniques for dealing with multiple valued functions" and generalizes Almgren's $Q$-valued functions. We study some relevant properties of such $Q$-multisections and apply the theory to provide an elementary and purely geometric proof of a delicate reparametrization theorem for multi-valued graphs which plays an important role in the regularity theory for higher codimension area minimizing currents \`a la Almgren-De Lellis-Spadaro.