Rectifiability of the singular set of multiple-valued energy minimizing harmonic maps

Hirsch, Jonas; Stuvard, Salvatore; Valtorta, Daniele

doi:10.1090/tran/7595

Cited by 4 publications

(5 citation statements)

References 30 publications

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“…The methods have some similar ingredients as those in Sections 5.3 and 5.4, but Lin gives independent proofs relying on facts at hand. De Lellis, Marchese, Spadaro and Valtorta [DMSV18] and Hirsch, Stuvard and Valtorta [HSV19] proved for Almgren's multivalued functions results analogous to Theorem 16.6. Alper proved in [Alp18] that the zero sets of harmonic maps from three-dimensional domains into a cone over the real projective plane are 1-rectifiable.…”

Section: Energy Minimizing Mapsmentioning

confidence: 77%

Rectifiability; a survey

Mattila¹

2021

Preprint

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Section: Energy Minimizing Mapsmentioning

confidence: 77%

Rectifiability; a survey

Mattila¹

2021

Preprint

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“…In this article, we adopt the same technique used in [13]: after giving an appropriate definition of singular stratification (adapted to the case of p-minimizing maps), we exploit the very same version of Reifenberg Theorem developed in [13,Theorems 3.3 and 3.4] to build a controlled covering of each stratum. Notice that analogous results (still exploiting this technique) are available for approximate harmonic maps [14] and for Q-valued energy minimizers [7].…”

Section: Introductionmentioning

confidence: 89%

“…Remark Notice that this definition is slightly different from the one used in [15] and originally introduced in [2]: indeed, in those papers u is defined to be "almost kinvariant" if it is (quantitatively) close in L p to a map which is 0-homogeneous and k-invariant. The two approaches can be shown to be equivalent (see for example [7,Proposition 3.11]); we won't explore this path further.…”

Section: Quantitative Stratificationsmentioning

confidence: 99%

Quantitative Regularity for p-Minimizing Maps Through a Reifenberg Theorem

Vedovato

2021

J Geom Anal

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In this article we extend to arbitrary p-energy minimizing maps between Riemannian manifolds a regularity result which is known to hold in the case $$p=2$$ p = 2 . We first show that the set of singular points of such a map can be quantitatively stratified: we classify singular points based on the number of almost-symmetries of the map around them, as done in Cheeger and Naber (Commun Pure Appl Math 66(6): 965–990, 2013). Then, adapting the work of Naber and Valtorta (Ann Math (2) 185(1): 131–227, 2017), we apply a Reifenberg-type Theorem to each quantitative stratum; through this, we achieve an upper bound on the Minkowski content of the singular set, and we prove it is k-rectifiable for a k which only depends on p and the dimension of the domain.

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“…A quick tutorial on the theory of multiple valued functions is contained in § 1.2, in order to ease the reading of the remaining part of the paper. As a byproduct, the theory of multiple valued Jacobi fields will show that the regularity theory for Dir-minimizing Q-valued functions is robust enough to allow one to produce analogous regularity results for minimizers of functionals defined on Sobolev spaces of Q-valued functions other than the Dirichlet energy (see also [DLFS11] for a discussion about general integral functionals defined on spaces of multiple valued functions and their semi-continuity properties, and [Hir16b,HSV17] for a regularity theory for multiple valued energy minimizing maps with values into a Riemannian manifold). 0.1.…”

Section: Introductionmentioning

confidence: 99%

Multiple valued Jacobi fields

Stuvard

2019

Calc. Var.

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We develop a multivalued theory for the stability operator of (a constant multiple of) a minimally immersed submanifold Σ of a Riemannian manifold M. We define the multiple valued counterpart of the classical Jacobi fields as the minimizers of the second variation functional defined on a Sobolev space of multiple valued sections of the normal bundle of Σ in M, and we study existence and regularity of such minimizers. Finally, we prove that any Q-valued Jacobi field can be written as the superposition of Q classical Jacobi fields everywhere except for a relatively closed singular set having codimension at least two in the domain. Assumption 1.1. We will consider:(M) a closed (i.e. compact with empty boundary) Riemannian manifold M of dimension m + k and class C 3,β for some β ∈ (0, 1); (S) a compact oriented minimal submanifold Σ of the ambient manifold M of dimension dim(Σ) = m and class C 3,β . Without loss of generality, we will also regard M as an isometrically embedded submanifold of some Euclidean space R d . We will let n := d − m and K := d − (m + k) be the codimensions of Σ and M in R d respectively.Let Σ m → M m+k ⊂ R d be as in Assumption 1.1. The Euclidean scalar product in R d is denoted ·, · . The metric on M and Σ is induced by the flat metric in R d : therefore, the same symbol will also denote the scalar product between tangent vectors to M or to Σ.The tangent space to M at a point z will be denoted T z M. The maps p M z : R d → T z M and p M⊥ z : R d → T ⊥ z M denote orthogonal projections of R d onto the tangent space to M at

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Rectifiability of the singular set of multiple-valued energy minimizing harmonic maps

Cited by 4 publications

References 30 publications

Rectifiability; a survey

Rectifiability; a survey

Quantitative Regularity for p-Minimizing Maps Through a Reifenberg Theorem

Multiple valued Jacobi fields

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