Regularity of area minimizing currents II: center manifold

Lellis, Camillo De; Spadaro, Emanuele

doi:10.4007/annals.2016.183.2.2

Cited by 51 publications

(248 citation statements)

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“…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: 99%

“…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%

“…[3, Section 3.4]), we introduce a variant of the frequency function and prove its almost monotonicity and boundedness. This analysis, carried in Sections 3, 4 and 5, relies on the variational formulas for images of multiple valued maps as computed in [5] and on the careful estimates of [6]. Our frequency function differs from that of Almgren and allows for simpler estimates.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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 ].…”

Section: Introductionmentioning

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

confidence: 99%

Section: Finite Order Of Contactmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Regularity of area minimizing currents III: blow-up

2016

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Uniqueness of Tangent Cones for Two‐Dimensional Almost‐Minimizing Currents

Lellis

Spadaro

Spolaor

2017

Comm Pure Appl Math

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We consider two‐dimensional integer rectifiable currents that are almost area minimizing and show that their tangent cones are everywhere unique. Our argument unifies a few uniqueness theorems of the same flavor, which are all obtained by a suitable modification of White's original theorem for area‐minimizing currents in the euclidean space. This note is also the first step in a regularity program for semicalibrated two‐dimensional currents and spherical cross sections of three‐dimensional area‐minimizing cones.© 2017 Wiley Periodicals, Inc.

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