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

Camillo De Lellis and Emanuele Spadaro

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Submission date: 05. Jun. 2013
Pages: 45
published in: Geometric and functional analysis, 24 (2014) 6, p. 1831-1884 
DOI number (of the published article): 10.1007/s00039-014-0306-3
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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 in 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.