# \(Q\)-Valued Functions Revisited

Share this page
*Camillo De Lellis; Emanuele Nunzio Spadaro*

In this memoir the authors revisit Almgren's theory of
\(Q\)-valued functions, which
are functions taking values in the space
\(\mathcal{A}_Q(\mathbb{R}^{n})\)
of unordered \(Q\)-tuples of points in \(\mathbb{R}^{n}\).

In particular, the authors:

- give shorter versions of Almgren's proofs of the existence of \(\mathrm{Dir}\)-minimizing \(Q\)-valued functions, of their Hölder regularity, and of the dimension estimate of their singular set;
- propose an alternative, intrinsic approach to these results, not relying on Almgren's biLipschitz embedding \(\xi: \mathcal{A}_Q(\mathbb{R}^{n})\to\mathbb{R}^{N(Q,n)}\);
- improve upon the estimate of the singular set of planar \(\mathrm{D}\)-minimizing functions by showing that it consists of isolated points.