REGULARITY OF AREA MINIMIZING CURRENTS I: GRADIENT Lp ESTIMATES

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 cur-rent, 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. 0. Foreword: a new proof of Almgren’s partial regularity In the present work we continue the investigations started in [14, 17], which together with the forthcoming papers [15, 16] lead to a proof of the following theorem. Theorem 0.1. Let Σ ⊂ Rm+n be a C3,ε0 submanifold for some ε0> 0 and T an m-dimensional area minimizing integral current in Σ. Then, there is a closed set Sing(T) of Hausdorff dimension at most m − 2 such that T is a C3,ε0 embedded submanifold in Σ \ (spt(∂T) ∪ Sing(T))