Regularity theory for 2-dimensional almost minimal currents I: Lipschitz approximation

Abstract

We construct Lipschitz Q Q -valued functions which carefully approximate 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 2 -dimensional integral currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of 3 3 -dimensional area minimizing cones.