Regularity of area minimizing currents III: blow-up

Abstract

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...

Regularity of area minimizing currents I: gradient L p estimates

“…We follow the proof of [10, Lemma 1.6] given in [10,Section 4]. First of all we notice that, once (16.10) and (16.11) are established, (16.12) follows from Theorem 17.1, since we clearly have that E no (T, C 11 √ m/2 , π 0 ) ≤ CE no (T 0 , B 6 √ m , π 0 ). Moreover, recall that there is a set of full measure A ⊂ B 5 √ m such that T, p π 0 , x is an integer rectifiable current for every x ∈ A.…”

Regularity Theory for 2-Dimensional Almost Minimal Currents II: Branched Center Manifold

“…However, all the maps constructed in this paper and used in the subsequent note [10] will approximate T with a high degree of accuracy in each Whitney region: such estimates are detailed in the next theorem. In order to simplify the notation, we will use …”

“…Thus a notation more consistent with that of [10] would be, in case (a) and (c), E Σ (T, B r (x)). However, the difference is a minor one and we prefer to keep our notation simpler.…”

Regularity of stable minimal hypersurfaces: Recent advances in the theory and applications

“…Furthermore, the result in this case is already sharp in view of the many two dimensional examples of oriented area minimizers provided by singular complex analytic varieties (such as {(z, w) : zw = 0} ∩ C × C or {(z, w) : z 2 = w 3 } ∩ C × C) which are all calibrated and consequently locally area minimizing. (See also the recent papers of De Lellis and Spadaro [DS,DS11] which present Almgren's lengthy proof concisely in a more accessible, step-by-step form, making also interesting connections between some of Almgren's key estimates and ideas in metric geometry and PDE).…”