Regularity of area minimizing currents I: gradient L^p estimates

TL;DR

This paper introduces a new a priori estimate for area minimizing currents, improving regularity assumptions and providing tools for better approximation via Lipschitz multiple valued graphs, advancing the understanding of geometric measure theory.

Contribution

It presents a novel higher integrability a priori estimate for area minimizing currents, enhancing the regularity theory and approximation techniques in geometric measure theory.

Findings

New a priori estimate on excess measure

Improved regularity assumptions for currents

Enhanced Lipschitz approximation methods

Abstract

In a series of papers, including the present one, we give a new, shorter proof of Almgren's partial regularity theorem for area minimizing currents in a Riemannian manifold, with a slight improvement on the regularity assumption for the latter. This note establishes a new a priori estimate on the excess measure of an area minimizing current, together with several statements concerning approximations with Lipschitz multiple valued graphs. Our new a priori estimate is an higher integrability type result, which has a counterpart in the theory of Dir-minimizing multiple valued functions and plays a key role in estimating the accuracy of the Lipschitz approximations.