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.
Foreword: a new proof of Almgren's partial regularityIn 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 ⊂ R m+n be a C 3,ε 0 submanifold for some ε 0 > 0 and T an mdimensional 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 C 3,ε 0 embedded submanifold in Σ \ (spt(∂T ) ∪ Sing(T )).Theorem 0.1 was first proved by Almgren in his monumental work [3], assuming slightly better regularity on Σ, namely Σ ∈ C 5 . The improvement itself is therefore not so significant, but our proof, besides being much shorter, introduces new ideas and establishes several new results, which we hope will provide useful tools for further investigations in the area. Indeed, although we still follow Almgren's program and use many of his groundbreaking discoveries, the main steps are achieved in a more efficient way thanks to new estimates and techniques. A striking example is the construction of the so-called center manifold, which is by far the most intricate part of Almgren's work and the least explored, in spite of its importance: in this respect, our construction in [15] is considerably simpler and shorter than [3, Chapter 4], and establishes better results.Some of our improvements are more transparent, although not substantially simpler, when Σ = R m+n and in a book in preparation [12] we will provide a complete and selfcontained account of Theorem 0.1 under such assumption. Moreover, building on our understanding of the various issues involved to the analysis of higher codimension singularities, we plan to tackle Chang's improvement [8], which shows that Sing(T ) consists of isolated points when m = 2. His arguments rely on a center manifold construction which does not match exactly the statements of [3] and it is not fully justified, but only briefly sketched in the appendix of [8]. In [18], instead, we give a detailed, simple construction for such center manifold and a complete proof of this refined regularity result.An alternative route to Chang's result for J-holomorphic currents in symplectic manifolds has been given recently in [27,28]. The interest in the regularity theory for this class of area minimizing 2-dimensional currents has been generated by t...