Inserted: 21 aug 2015
Last Updated: 8 jun 2016
To appear in Transactions of the AMS
We construct Lipschitz $Q$-valued functions which approximate carefully integral currents when their cylindrical excess is small and they are almost minimizing in a suitable sense. This result is used in two subsequent works to prove the discreteness of the singular set for the following three classes of $2$-dimensional integral currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of $3$-dimensional area minimizing cones.
Keywords: regularity, integral currents, area minimizing, semicalibrations