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Regularity of area minimizing currents I: gradient L p estimates

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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 a 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.

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De Lellis, C., Spadaro, E. Regularity of area minimizing currents I: gradient L p estimates. Geom. Funct. Anal. 24, 1831–1884 (2014).

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