Analyse des problèmes conformément invariants

Laurain, Paul

doi:10.5802/slsedp.110

Cited by 3 publications

(2 citation statements)

References 39 publications

(30 reference statements)

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“…Added in proof: We were informed that Laurain and Petrides have recently proved a similar result, and Theorem 2.2 was announced in a seminar proceedings paper [26] but without a proof.…”

Section: Acknowledgementsmentioning

confidence: 99%

Min-max minimal disks with free boundary in Riemannian manifolds

2020

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In this paper, we establish a min-max theory for constructing minimal disks with free boundary in any closed Riemannian manifold. The main result is an effective version of the partial Morse theory for minimal disks with free boundary established by Fraser. Our theory also includes as a special case the min-max theory for Plateau problem of minimal disks, which can be used to generalize the famous work by Morse-Thompkins and Shiffman on minimal surfaces in R n to the Riemannian setting.More precisely, we generalize the min-max construction of minimal surfaces using harmonic replacement introduced by Colding and Minicozzi to the free boundary setting. As a key ingredient to this construction, we show an energy convexity for weakly harmonic maps with mixed Dirichlet and free boundaries from the half unit 2-disk in R 2 into any closed Riemannian manifold, which in particular yields the uniqueness of such weakly harmonic maps. This is a free boundary analogue of the energy convexity and uniqueness for weakly harmonic maps with Dirichlet boundary on the unit 2-disk proved by Colding and Minicozzi.

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“…Added in proof: We were informed that Laurain and Petrides have recently proved a similar result, and Theorem 2.2 was announced in a seminar proceedings paper [26] but without a proof.…”

Section: Acknowledgementsmentioning

confidence: 99%

Min-max minimal disks with free boundary in Riemannian manifolds

2020

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“…In fact, the crucial point is the compactness compensation phenomena which appears in all conformally invariant problems. This remark permits the first author and Lin [25] to prove the convexity for some biharmonic maps, see also [24] for a review. In the present paper we prove the following free boundary version of Colding-Minicozzi energy convexity.…”

Section: Introductionmentioning

confidence: 91%

Existence of min-max free boundary disks realizing the width of a manifold

Laurain

Petrides

2019

Advances in Mathematics

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Energy convexity of intrinsic bi-harmonic maps and applications I: Spherical target

Laurain

Lin

2020

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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In this paper, we show an energy convexity and thus uniqueness for weakly intrinsic bi-harmonic maps from the unit 4-ball {B_{1}\subset\mathbb{R}^{4}} into the sphere {\mathbb{S}^{n}}. In particular, this yields a version of uniqueness of weakly harmonic maps on the unit 4-ball which is new. We also show a version of energy convexity along the intrinsic bi-harmonic map heat flow into {\mathbb{S}^{n}}, which in particular yields the long-time existence of the intrinsic bi-harmonic map heat flow, a result that was until now only known assuming the non-positivity of the target manifolds by Lamm [26]. Further, we establish the previously unknown result that the energy convexity along the flow yields uniform convergence of the flow.

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Analyse des problèmes conformément invariants

Abstract: Analyse des problèmes conformément invariantsSéminaire Laurent Schwartz -EDP et applications (2016

Cited by 3 publications

References 39 publications

Min-max minimal disks with free boundary in Riemannian manifolds

Min-max minimal disks with free boundary in Riemannian manifolds

Existence of min-max free boundary disks realizing the width of a manifold

Energy convexity of intrinsic bi-harmonic maps and applications I: Spherical target

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