Regularity theory for a class of 2-dimensional almost area minimizing currents

Abstract

In this thesis we deal with interior regularity issues for area minimizing surfaces. In particular, we consider a special class of almost area minimizing, 2-dimensional integral currents, with bounded mean curvature, and we prove that their interior singular set is discrete. More specifically, we treat area minimzing currents in riemannian manifolds, semicalibrated currents and spherical cross sections of 3-dimensional area minimizing cones. In all these three situations our result is sharp. Moreover, a nice corollary of our theorem is the fact that the singular set of 3-dimensional area minimizing cones consists of at most a finite number of lines.
Our result is inspired by the approach of Almgren-Chang (cf. [9]) for area minimizing currents, which we revisit and complete, adding also some new cases. In particular we use a lot of techniques coming from De Lellis and Spadaro’s new proof of Almgren’s Big Regularity paper (cf. [17, 18, 19, 20, 21]). Other important known results that we manage to cover are Tian-Riviére regularity theorem for almost complex curves (cf. [53]) and Bellettini- Riviére extension to a class of semicalibrated 3-dimensional cones (cf. [7]). Our result for general semicalibrated currents and general 3-dimensional area minimizing cones is entirely new.
It is worth mentioning that, among the various steps in the proof of our main result, we give a unified and much shorter proof of already existing results concerning the uniqueness of the tangent cone to 2-dimensional area minimizing and semicalibrated currents (cf. [66, 46]), generalizing it to the larger class of almost area minimizing 2-dimensional currents. This is done relying heavily on [66]. Moreover, we also generalize a Lipschitz approximation result for area minimizing currents, proved first by Almgren (cf. [3]) and recently revisited by De Lellis and Spadaro (cf. [19]). In particular this result is independent from the dimension of the current.
The other two fundamental tools are the so called Center Manifold and the Frequency function, for which we were inspired by [3, 9, 20, 21], and which we combine in an inductive
argument to conclude our main theorem.
All the results of this thesis where obtained in collaboration with Camillo De Lellis and Emanuele Spadaro, to whom I deeply grateful for guiding me step by step in the beautiful (and hard) world of geometric measure theory.