*Published Paper*

**Inserted:** 29 feb 2008

**Last Updated:** 24 may 2017

**Journal:** Memoirs of the American Mathematical Society

**Volume:** 991

**Year:** 2011

**Abstract:**

In this note we revisit Almgren's theory of Q-valued functions, that are functions taking values in the space $\mathcal{A}_Q (*R*^n)$ of unordered Q-tuples of points in $*R*^n$. In particular: \begin{itemize} \item we give shorter versions of Almgren's proofs of the existence of Dir-minimizing Q-valued functions, of their HÃ¶lder regularity and of the dimension estimate of their singular set; \item we propose an alternative intrinsic approach to these results, not relying on Almgren's biLipschitz embedding of $\mathcal{A}_Q (*R*^n)$ in the euclidean space; \item we improve upon the estimate of the singular set of planar Dir-minimizing functions by showing that it consists of isolated points.

For the most updated version and eventual errata see the page

http:/www.math.uzh.ch*index.php?id=publikationen&key1=493
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