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Abstract. This is the last of a series of three papers in which we give a new, shorter proof of a slightly improved version of Almgren's partial regularity of area minimizing currents in Riemannian manifolds. Here we perform a blow-up analysis deducing the regularity of area minimizing currents from that of Dir-minimizing multiple valued functions. IntroductionIn this paper we complete the proof of a slightly improved version of the celebrated Almgren's partial regularity result for area minimizing currents in a Riemannian manifold (see [1]), namely Theorem 0.3 below.In this paper we follow the notation of [6] concerning balls, cylinders and disks. In particular B r (x) ⊂ R m+n will denote the Euclidean ball of radius r and center x.Definition 0.2. For T and Σ as in Assumption 0.1 we defineThe partial regularity result proven first by Almgren [1] under the more restrictive hypothesis Σ ∈ C 5 gives an estimate on the Hausdorff dimension dim H (Sing(T )) of Sing(T ).Theorem 0.3. dim H (Sing(T )) ≤ m − 2 for any m,n, l, T and Σ as in Assumption 0.1.In this note we complete the proof of Theorem 0.3, based on our previous works [3,5,4,6], thus providing a new, and much shorter, account of one of the most fundamental regularity result in geometric measure theory; we refer to [4] for an extended general introduction to all these works. The proof is carried by contradiction: in the sequel we will always assume the following.Assumption 0.4 (Contradiction). There exist m ≥ 2,n, l, Σ and T as in Assumption 0.1 such that H m−2+α (Sing(T )) > 0 for some α > 0. The hypothesis m ≥ 2 in Assumption 0.4 is justified by the well-known fact that Sing(T ) = ∅ when m = 1 (in this case spt(T ) \ spt(∂T ) is locally the union of finitely many non-intersecting geodesic segments). Starting from Assumption 0.4, we make a careful blow-up analysis, split in the following steps. 0.1. Flat tangent planes. We first reduce to flat blow-ups around a given point, which in the sequel is assumed to be the origin. These blow-ups will also be chosen so that the size of the singular set satisfies a uniform estimate from below (cf. Section 1). 0.2. Intervals of flattening. For appropriate rescalings of the current around the origin we take advantage of the center manifold constructed in [6], which gives a good approximation of the average of the sheets of the current at some given scale. However, since it might fail to do so at different scales, in Section 2 we introduce a stopping condition for the center manifolds and define appropriate intervals of flattening I j = [s j , t j ]. For each j we construct a different center manifold M j and approximate the (rescaled) current with a suitable multi-valued map on the normal bundle of M j . 0.3. Finite order of contact. A major difficulty in the analysis is to prove that the minimizing current has finite order of contact with the center manifold. To this aim, in analogy with the case of harmonic multiple valued functions (cf. [3, Section 3.4]), we introduce a variant of the frequency function and prove its...

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...

We prove several results on Almgren's multiple valued functions and their links to integral currents. In particular, we give a simple proof of the fact that a Lipschitz multiple valued map naturally defines an integer rectifiable current; we derive explicit formulae for the boundary, the mass and the first variations along certain specific vector-fields; and exploit this connection to derive a delicate reparametrization property for multiple valued functions. These results play a crucial role in our new proof of the partial regularity of area minimizing currents [5][6][7].

Abstract. We give a detailed description of the geometry of single droplet patterns in a nonlocal isoperimetric problem. In particular we focus on the sharp interface limit of the OhtaKawasaki free energy for diblock copolymers, regarded as a paradigm for those energies modeling physical systems characterized by a competition between short and a long-range interactions. Exploiting fine properties of the regularity theory for minimal surfaces, we extend previous partial results in different directions and give robust tools for the geometric analysis of more complex patterns. IntroductionIn several physical systems competing short-range attractive and long-range repulsive interactions often lead to the formation of mesoscopic scale patterns. Roughly speaking, the short-range interactions favor phase-separation on a microscopic scale, while the long-range ones frustrate such an ordering on the scale of the whole sample. When these systems can be described in terms of a free energy, such a phenomenon is usually referred to as an energy-driven pattern formation. Examples of energy-driven patterns are ubiquitous in physics: among the others we recall ferromagnetic and polymeric systems, type-I superconductor films and Langmuir layers. Even if these systems are driven by different physical laws, they exhibit remarkable similarities in the overall geometry of the formed patterns (see [33] and [46]).Our principal interest is the description of the geometry of patterns. For this reason, we leave for further studies the detailed analysis of more realistic systems and we focus here on an energy model which encodes only the main features of pattern-formation. More specifically, in what follows we are interested in the minimization of the following energy functional:where u is the order parameter of a two-phases system confined in Ω ⊂ R n , and γ and m are two nonnegative numerical parameters. The two terms in the energy mimic attractive short-range and repulsive long-range energies between the phases. More precisely the first term is local, it favors minimal interface area and drives the system toward a partition into few pure phases while the second term, involving a Coulomb-like kernel G, is non-local and favor a fine mixing of the phases. A detailed description of the energy is given in § 1. The competition between these two terms is expected to induce the formation of highly regular mesoscopic patterns, whose geometry strongly depends on the choice of the parameters γ and m (e.g. spherical spots, cylinders, gyroids and lamellae). 0.1. The Ohta-Kawasaki functional for diblock copolymers. The model we consider arises as a simplification of a Ginzburg-Landau functional proposed by Ohta and Kawasaki in their pioneering paper [41] as a possible description of a diblock copolymers' (DBC) system. Even though it is questionable whether such an energy actually describes DBCs (see Choksi and Ren [15], Muratov [39] and Niethammer and Oshita [40]), nevertheless it is a first, and mathematically non-trivial, attempt to capture some...

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Building upon the recent results in [15] we provide a thorough description of the free boundary for solutions to the fractional obstacle problem in R n+1 with obstacle function ϕ (suitably smooth and decaying fast at infinity) up to sets of null H n−1 measure. In particular, if ϕ is analytic, the problem reduces to the zero obstacle case dealt with in [15] and therefore we retrieve the same results:(i) local finiteness of the (n − 1)-dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure), (ii) H n−1 -rectifiability of the free boundary, (iii) classification of the frequencies and of the blow-ups up to a set of Hausdorff dimension at most (n − 2) in the free boundary. Instead, if ϕ ∈ C k+1 (R n ), k ≥ 2, similar results hold only for distinguished subsets of points in the free boundary where the order of contact of the solution with the obstacle function ϕ is less than k + 1.2010 Mathematics Subject Classification. Primary 35R35, 49Q20.

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