$Q$-Valued Functions Revisited

$Q$-Valued Functions Revisited Synopsis

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 Hoelder 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 $xi: mathcal{A}_Q(mathbb{R}^{n}) omathbb{R}^{N(Q,n)}$; improve upon the estimate of the singular set of planar $mathrm{D}$-minimizing functions by showing that it consists of isolated points.