Regularity of area minimizing currents I: gradient L p estimates

“…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.…”

Section: Sing(t ) := Spt(t ) \ Spt(∂t ) ∪ Reg(t ) (02)mentioning

confidence: 89%

“…The two subsequences might, however, differ: in the next proposition we show the existence of one point and a single subsequence along which both conclusions hold. For the relevant notation (concerning, for instance, excess and height of currents) we refer to [4,6]. …”

Section: Finite Order Of Contactmentioning

confidence: 99%

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Regularity of area minimizing currents III: blow-up

2016

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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...

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“…Basic notation and first main assumptions. For the notation concerning submanifolds Σ ⊂ R 2+n we refer to [7,Section 1]. With B r (p) and B r (x) we denote, respectively, the open ball with radius r and center p in R 2+n and the open ball with radius r and center x in R 2 .…”

Section: ])mentioning

confidence: 99%

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

2017

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Abstract. We construct a branched center manifold in a neighborhood of a singular point of a 2-dimensional integral current which is almost minimizing in a suitable sense. Our construction is the first half of an argument which shows the discreteness of the singular set for the following three classes of 2-dimensional currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of 3-dimensional area minimizing cones.This paper is the third in a series of works aimed at establishing an optimal regularity theory for 2-dimensional integral currents which are almost minimizing in a suitable sense. Building upon the monumental work of Almgren [1], Chang in [4] established that 2-dimensional area minimizing currents in Riemannian manifolds are classical minimal surfaces, namely they are regular (in the interior) except for a discrete set of branching singularities. The argument of Chang is however not entirely complete since a key starting point of his analysis, the existence of the so-called "branched center manifold", is only sketched in the appendix of [4] and requires the understanding (and a suitable modification) of the most involved portion of the monograph [1].An alternative proof of Chang's theorem has been found by Rivière and Tian in [16] for the special case of J-holomorphic curves. Later on the approach of Rivière and Tian has been generalized by Bellettini and Rivière in [3] to handle a case which is not covered by [4], namely that of special Legendrian cycles in S 5 (see also [2] for a further generalization). Meanwhile the first and second author revisited Almgren's theory giving a much shorter version of his program for proving that area minimizing currents are regular up to a set of Hausdorff codimension 2, cf. [6,8,7,9,10]. In this note and its companion papers [11,12] we build upon the latter works in order to give a complete regularity theory which includes both the theorems of Chang and Bellettini-Rivière as special cases. In order to be more precise, we introduce the following terminology (cf. [13, Definition 0.3 ]).Definition 0.1. Let Σ ⊂ R m+n be a C 2 submanifold and U ⊂ R m+n an open set.(a) An m-dimensional integral current T with finite mass and spt(T ) ⊂ Σ ∩ U is area minimizing in Σ ∩ U if M(T + ∂S) ≥ M(T ) for any m + 1-dimensional integral current S with spt(S) ⊂⊂ Σ ∩ U . (b) A semicalibration (in Σ) is a C 1 m-form ω on Σ such that ω x c ≤ 1 at every x ∈ Σ, where · c denotes the comass norm on Λ m T x Σ. An m-dimensional integral current T with spt(T ) ⊂ Σ is semicalibrated by ω if ω x ( T ) = 1 for T -a.e. x. In what follows, given an integer rectifiable current T , we denote by Reg(T ) the subset of spt(T ) \ spt(∂T ) consisting of those points x for which there is a neighborhood U such that T U is a (constant multiple of) a C 2 submanifold. Correspondingly, Sing(T ) is the set spt(T ) \ (spt(∂T ) ∪ Reg(T )). Observe that Reg(T ) is relatively open in spt(T ) \ spt(∂T ) and thus Sing(T ) is relatively closed. The main result of this and the works ...

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Regularity of area minimizing currents I: gradient L p estimates

Cited by 61 publications

References 25 publications

Higher integrability of the gradient for minimizers of the 2 d Mumford–Shah energy

Higher integrability of the gradient for minimizers of the 2 d Mumford–Shah energy

Regularity of area minimizing currents III: blow-up

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

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