Multiple valued functions and integral currents (Q2792160)

In [Ann. Math. (2) 72, 458--520 (1960; Zbl 0187.31301)], \textit{H. Federer} and \textit{W. H. Fleming} studied normal and integral currents. A current \(T\), in the sense of de Rham, is normal if it has a compact support and if both \(T\) and its boundary are bounded, and it is called rectifiable if it is the limit in norm of a finite polyhedral chain with integer coefficients. A current \(T\) which is both normal and rectifiable is integral. In his fundamental work [``Almgren's big regularity paper. Q-valued functions minimizing Dirichlet's integral and the regularity of area-minimizing rectifiable currents up to codimension 2. Edited by V. Scheffer and Jean E. Taylor'', World Scientific Monograph Series in Mathematics 1. Singapore: World Scientific (2000; Zbl 0985.49001)], \textit{F. J. Almgren} considered the theory of maps \({f\:\Omega\to {\mathcal A}_Q({\mathbb R}^n)}\), where \(\Omega\subset{\mathbb R}^m\) is an open set and \({\mathcal A}_Q({\mathbb R}^n)\) is the set of unordered \(Q\)-tuples of points in \({\mathbb R}^n\), and announced his famous regularity result stating that an \(m\)-dimensional mass minimizing integral current is regular away from its boundary except for a closed subset of dimension at most \(m-2\). Let \(f:\mathbb R^m\supset\Omega\to\mathbb R^n\) be a Lipschitz map and \(\text{Gr}(f)\) its graph. If the Lipschitz constant of \(f\) is small and the coordinates in \(\mathbb R^{m+n}\) are changed with an orthogonal transformation close to the identity, then the set \(\text{Gr}(f)\) is also the graph of a Lipschitz function \(\tilde f\) over some domain \(\widetilde\Omega\) in the new coordinate system. There exist suitable Lipschitz maps \(\Psi\) and \(\Phi\) such that \(\tilde f(x)=\Psi(x,f(\Phi(x)))\). In the multiple valued case, \(\text{Gr}(f)\) is still the graph of a new Lipschitz map \(\tilde f\) in the new coordinate system. The connection between multiple valued functions and integral currents is crucial in the analysis of the regularity of area minimizing currents. It provides the necessary tools for the approximation of currents with graphs of multiple valued function.NEWLINENEWLINEIn this paper, the authors prove several results on Almgren's multiple valued functions and their links to integral currents. In particular, they give a simple proof of the fact that a Lipschitz multiple valued map naturally defines an integer rectifiable current, and derive explicit formulas for the boundary. If \(\Sigma\) is a Lipschitz submanifold of \(\mathbb R^N\) with Lipschitz boundary, \(F:\Sigma\to\mathcal{A}_Q(\mathbb R^n)\) is a proper Lipschitz function, and \(f=F_{|\partial\Sigma}\), then \(\partial\mathbf{T}_F=\mathbf{T}_f\), where \(\mathbf{T}_F\) is the current associated to the graph \(\text{Gr}(f)\). Also, the authors provide Taylor expansions for those various boundary formulas when the current is sufficiently flat.