Q-valued functions revisited

TL;DR

This paper revisits Almgren's theory of Q-valued functions, providing shorter proofs, an intrinsic approach, and improved singular set estimates, particularly showing isolated singularities in the planar case.

Contribution

It offers more concise proofs, an intrinsic methodology avoiding Almgren's embedding, and refined singular set estimates for Q-valued Dir-minimizing functions.

Findings

Shorter proofs of key existence and regularity results

An intrinsic approach to Q-valued functions

Singular set in planar case consists of isolated points

Abstract

In this note we revisit Almgren's theory of Q-valued functions, that are functions taking values in the space of unordered Q-tuples of points in R^n. In particular: 1) we give shorter versions of Almgren's proofs of the existence of Dir-minimizing Q-valued functions, of their Hoelder regularity and of the dimension estimate of their singular set; 2) we propose an alternative intrinsic approach to these results, not relying on Almgren's biLipschitz embedding; 3) we improve upon the estimate of the singular set of planar Dir-minimizing functions by showing that it consists of isolated points.