Rigidity of minimal Lagrangian diffeomorphisms between spherical cone surfaces

Christian El Emam; Andrea Seppi

doi:10.5802/jep.190

Rigidity of minimal Lagrangian diffeomorphisms between spherical cone surfaces
[Rigidité des difféomorphismes minimaux lagrangiens entre surfaces sphériques à singularités coniques]

Christian El Emam ¹ ; Andrea Seppi ²

¹ Université du Luxembourg, Maison du Nombre, 6 Avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg
² Institut Fourier, UMR 5582, Laboratoire de Mathématiques, Université Grenoble Alpes CS 40700, 38058 Grenoble cedex 9, France

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 581-600.

Résumé
Abstract

Nous démontrons que toute application minimale lagrangienne entre deux surfaces fermées sphériques à singularités coniques est une isométrie, sans aucune hypothèse sur les valeurs des multi-angles des deux surfaces. En appliquant ce résultat, nous prouvons une généralisation du théorème classique de rigidité de Liebmann, notamment l’énoncé que toute immersion dans l’espace euclidien de dimension $3$ d’une surface fermée avec courbure gaussienne constante positive et avec points de ramification est un revêtement ramifié sur une sphère.

We prove that any minimal Lagrangian diffeomorphism between two closed spherical surfaces with cone singularities is an isometry, without any assumption on the multiangles of the two surfaces. As an application, we show that every branched immersion of a closed surface of constant positive Gaussian curvature in Euclidean three-space is a branched covering onto a round sphere, thus generalizing the classical rigidity theorem of Liebmann to branched immersions.

Reçu le : 2021-01-18
Accepté le : 2022-02-28
Publié le : 2022-03-10

DOI : 10.5802/jep.190

Classification : 53C24, 53A05, 53C42, 53C43
Keywords: Spherical surfaces, conical singularities, minimal Lagrangian maps, immersions in Euclidean space, isolated singularities, Gaussian curvature
Mot clés : Surfaces sphériques, singularités coniques, applications minimales lagrangiennes, immersions dans l’espace euclidien, singularités isolées, courbure gaussienne

Affiliations des auteurs :

Christian El Emam ¹ ; Andrea Seppi ²

¹ Université du Luxembourg, Maison du Nombre, 6 Avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg
² Institut Fourier, UMR 5582, Laboratoire de Mathématiques, Université Grenoble Alpes CS 40700, 38058 Grenoble cedex 9, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Christian El Emam and Andrea Seppi},
     title = {Rigidity of minimal {Lagrangian} diffeomorphisms between spherical cone surfaces},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {581--600},
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     year = {2022},
     doi = {10.5802/jep.190},
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Christian El Emam; Andrea Seppi. Rigidity of minimal Lagrangian diffeomorphisms between spherical cone surfaces. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 581-600. doi : 10.5802/jep.190. https://jep.centre-mersenne.org/articles/10.5802/jep.190/

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