Multiple valued functions and integral currents

Lellis, Camillo De; Spadaro, Emanuele

doi:10.2422/2036-2145.201306_002

Cited by 30 publications

(111 citation statements)

References 11 publications

(37 reference statements)

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“…Note that X i is the infinitesimal generator of a one parameter family of bilipschitz homeomorphisms Φ ε defined as Φ ε (q) := Ψ ε (p(q)) + q − p(q), where Ψ ε is the oneparameter family of bilipschitz homeomorphisms of M generated by Y . Consider now the map F (q) := i q + N i (q) and the current T F associated to its image: in particular we are using the conventions of [DLS15] in the case Q < p 2 (i.e. when N takes values in A Q (R m+n )) and the conventions introduced in [DLHMS, Definition 11.2] in the case Q = p 2 (i.e.…”

Section: First Variationsmentioning

confidence: 99%

Regularity of area minimizing currents mod p

et al. 2020

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We establish a first general partial regularity theorem for area minimizing currents $${\mathrm{mod}}(p)$$ mod ( p ) , for every p, in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of an m-dimensional area minimizing current $${\mathrm{mod}}(p)$$ mod ( p ) cannot be larger than $$m-1$$ m - 1 . Additionally, we show that, when p is odd, the interior singular set is $$(m-1)$$ ( m - 1 ) -rectifiable with locally finite $$(m-1)$$ ( m - 1 ) -dimensional measure.

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Section: First Variationsmentioning

confidence: 99%

Regularity of area minimizing currents mod p

et al. 2020

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“…Moreover the above example can be easily modified to provide an example of a two dimensional mass minimizing current for which there exists a sequence of interior singular points accumulating towards the boundary. This shows that the (interior) regularity results for two dimensional mass minimizing currents in [8,12,14,13,15,16] are actually optimal, see Remark 2.2.…”

Section: Introductionmentioning

confidence: 77%

“…The latter papers are based on a revisitation of Almgren's theory, due to the first author and Emanuele Spadaro (cf. [12,14,13,15,16]), which simplifies Almgren's proof introducing several new ideas. The latter works are indeed one of the starting points of this paper.…”

Section: Introductionmentioning

confidence: 99%

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On the boundary behavior of mass-minimizing integral currents

Lellis¹,

Philippis²,

Hirsch³

et al. 2018

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Let Σ be a smooth Riemannian manifold, Γ ⊂ Σ a smooth closed oriented submanifold of codimension higher than 2 and T an integral area-minimizing current in Σ which bounds Γ. We prove that the set of regular points of T at the boundary is dense in Γ. Prior to our theorem the existence of any regular point was not known, except for some special choice of Σ and Γ. As a corollary of our theorem• we answer to a question of Almgren showing that, if Γ is connected, then T has at least one point p of multiplicity 1 2 , namely there is a neighborhood of the point p where T is a classical submanifold with boundary Γ;• we generalize Almgren's connectivity theorem showing that the support of T is always connected if Γ is connected; • we conclude a structural result on T when Γ consists of more than one connected component, generalizing a previous theorem proved by Hardt and Simon when Σ = R m+1 and T is m-dimensional. ContentsChapter 1. Introduction Chapter 2. Corollaries, open problems and plan of the paper 2.1. Indecomposable components of T 2.2. Almgren's question and proof of Theorem 1.9 2.3. Proof of Theorem 1.8 2.4. Plan of the proof of Theorem 1.6 2.5. Open problems Chapter 3. Stratification and reduction to collapsed points 3.1. First variation and monotonicity formula 3.2. Stratification 3.3. Proof of Theorem 3.8 3.4. Proof of Theorem 3.2, of Corollary 3.11 and of Corollary 3.12 Chapter 4. Regularity for Q − 1 2 Dir-minimizers 4.1. Preliminaries and proof of Theorem 4.2 4.1.1. Interpolation lemma 4.1.2. A simple measure theoretical lemma 4.1.3. Proof of Theorem 4.8: Compactness 4.1.4. Proof of Theorem 4.8: Minimality 4.2. The main frequency function estimate 4.2.1. H and D 4.2.2. Lower bound on H 4.2.3. Outer variations 4.2.4. Inner variations 4.2.5. A good function d 4.2.6. Proof of Theorem 4.15 4.3. Further consequences of the frequency function estimate 4.4. Blowup: proof of Theorem 4.5 with ϕ ≡ 0 4.5. Proof of Theorem 4.5: general case Chapter 5. First Lipschitz approximation and harmonic blow-up 5.1. Proof of Theorem 5.5 5.1.1. Artificial sheet and "bad set" 5.1.2. Lipschitz estimate 9.4. First variations 9.4.1. Outer variation 9.4.2. Inner variation 9.5. Key identities 9.6. Estimates on the error terms 9.6.1. Families of subregions 9.6.2. Lower and upper bounds in the subregions 9.6.3. Estimates on the error terms 9.7. Proof of Theorem 9.3 Chapter 10. Final blow-up argument 10.1. Asymptotics for D(r) 10.2. Vanishing of the average 10.3. Minimality and convergence in energy Bibliography Index CHAPTER 1

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“…The standard references for multi-valued functions are [AJ00], [DL10], and [DLS13]; the reader wishing to attain a broader background on multi-valued functions is recommended to consult these, as we shall only recall the notions we need.…”

Section: Preliminariesmentioning

confidence: 99%

A Campanato Regularity Theory for Multi-Valued Functions with Applications to Minimal Surface Regularity Theory

Minter¹

2021

Preprint

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The regularity theory of the Campanato space L (q,λ) k(Ω) has been successfully used to understand the regularity of solutions to certain geometric variational problems. Here we extend this theory to multi-valued functions, adapting for the most part Campanato's original ideas in [Cam64]. We then give an application within the regularity theory of stationary integral varifolds. More precisely, we prove a regularity theorem for certain blow-up classes of functions, which typically arise when studying blow-ups of sequences of stationary integral varifolds converging to higher multiplicity planes or unions of half-planes. In such a setting, based in part on ideas in [Wic14] and [BKW21], we are able to deduce a boundary regularity theory for multi-valued harmonic functions, which is the first of its kind. In conjunction with [Min21], the results here establish a regularity theorem for stable codimension one integral varifolds near classical cones of density 5 2 .

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Multiple valued functions and integral currents

Cited by 30 publications

References 11 publications

Regularity of area minimizing currents mod p

Regularity of area minimizing currents mod p

On the boundary behavior of mass-minimizing integral currents

A Campanato Regularity Theory for Multi-Valued Functions with Applications to Minimal Surface Regularity Theory

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