A gap theorem for $α$-harmonic maps between two-spheres

Lamm, Tobias; Malchiodi, Andrea; Micallef, Mario

doi:10.48550/arxiv.1903.10217

Cited by 2 publications

(3 citation statements)

References 15 publications

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“…We have the following lemma 7 Unless the Ginzburg-Landau relaxation combined with the entropy estimate (III.9) is making a very special selection and is breaking the asymptotic Möbius group action. Such gauge breaking effect by relaxation has been already observed in [10,11] Lemma IV.1. Let G be a smooth map from the disc…”

supporting

confidence: 67%

Minmax Hierarchies, Minimal Fibrations and a PDE based Proof of the Willmore Conjecture

Rivière¹

2020

Preprint

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We introduce a general scheme that permits to generate successive min-max problems for producing critical points of higher and higher indices to Palais-Smale Functionals in normal Banach manifolds equipped with complete Finsler structures. We call the resulting tree of minmax problems a minmax hierarchy. We give several examples and in particular we explain how to implement this scheme in the framework of the viscosity method introduced by the author in [35] in order to give a new proof of the Willmore conjecture after the famous result by Marques and Neves.

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supporting

confidence: 67%

Minmax Hierarchies, Minimal Fibrations and a PDE based Proof of the Willmore Conjecture

Rivière¹

2020

Preprint

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“…We are not aware of results similar to Theorem 5.2 or Theorem 5.1 in the literature. However, it seems that a somewhat similar effect is underlying the arguments in the recent work by Lamm-Malchiodi-Micallef [60].…”

Section: Balanced Energy Estimate For the Non-scaling Invariant Normssupporting

confidence: 52%

“…We are not aware of such a statement in the literature even in the local case of p-harmonic maps, see Theorem 5.2. However, see [60], where the 3 It would be more in line with the original approach of Sacks-Uhlenbeck if we chose W s, n s α -minimizers, α → 1 + . However, that would have the technical drawback that W s, n s α ֒→ W s, n s loc for α > 1 and s ∈ (0, 1), see [78].…”

mentioning

confidence: 82%

Minimal $W^{s,\frac{n}{s}}$-harmonic maps in homotopy classes

Mazowiecka¹,

Schikorra²

2020

Preprint

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Let Σ a closed n-dimensional manifold, N ⊂ R M be a closed manifold, and u ∈ W s, n s (Σ, N ) for s ∈ (0, 1). We extend the monumental work of Sacks and Uhlenbeck by proving that if π n (N ) = {0} then there exists a minimizing W s, n s -harmonic map homotopic to u. If π n (N ) = {0}, then we prove that there exists a W s, n s -harmonic map from S n to N in a generating set of π n (N ).Since several techniques, especially Pohozaev-type arguments, are unknown in the fractional framework (in particular when n s = 2 one cannot argue via an extension method), we develop crucial new tools that are interesting on their own: such as a removability result for point-singularities and a balanced energy estimate for non-scaling invariant energies. Moreover, we prove the regularity theory for minimizing W s, n s -maps into manifolds. ContentsW s, n s -HARMONIC MAPS IN HOMOTOPY CLASSES 5• K.M.

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A gap theorem for $α$-harmonic maps between two-spheres

Cited by 2 publications

References 15 publications

Minmax Hierarchies, Minimal Fibrations and a PDE based Proof of the Willmore Conjecture

Minmax Hierarchies, Minimal Fibrations and a PDE based Proof of the Willmore Conjecture

Minimal $W^{s,\frac{n}{s}}$-harmonic maps in homotopy classes

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