Q-valued functions revisited

Source abstract

In this note we revisit Almgren’s theory of Q Q -valued functions, that are functions taking values in the space A Q ( R n ) \mathcal {A}_Q(\mathbb {R}^{n}) of unordered Q Q -tuples of points in R n \mathbb {R}^{n} . In particular: we give shorter versions of Almgren’s proofs of the existence of D i r \mathrm {Dir} -minimizing Q Q -valued functions, of their Hölder regularity and of the dimension estimate of their singular set; we propose an alternative, intrinsic approach to these results, not relying on Almgren’s biLipschitz embedding ξ : A Q ( R n ) → R N ( Q , n ) \xi : \mathcal {A}_Q(\mathbb {R}^{n})\to \mathbb {R}^{N(Q,n)} ; we improve upon the estimate of the singular set of planar D \mathrm {D} -minimizing functions by showing that it consists of isolated points.