Blow-up analysis of stationary two-valued functions (Q2509513)

The author introduces the notions of squeeze (resp. squash) stationary functions as multiple-valued functions (in the sense of \textit{F. J. Almgren jun.} [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. Singapore: World Scientific (2000; Zbl 0985.49001)]) stationary with respect to diffeomorphisms of the domain (resp. range). It is shown that two-valued two-dimensional squeeze stationary functions are locally Lipschitz continuous. This fact is used to prove the strong convergence of such functions to their unique blow-up at any point. The main result states that the branch set of such a function consists of finitely many real analytic curves meeting at nod points with equal angles.