Inserted: 2 jul 2013
Last Updated: 9 sep 2014
In a series of papers, including the present one, we give a new, shorter proof of Almgren's partial regularity theorem for area mi\-ni\-mi\-zing 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.