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 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.
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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
@article{JEP_2022__9__581_0, author = {El Emam, Christian and Seppi, Andrea}, title = {Rigidity of minimal {Lagrangian} diffeomorphisms between spherical cone surfaces}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {581--600}, publisher = {Ecole polytechnique}, volume = {9}, year = {2022}, doi = {10.5802/jep.190}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jep.190/} }
TY - JOUR AU - El Emam, Christian AU - Seppi, Andrea TI - Rigidity of minimal Lagrangian diffeomorphisms between spherical cone surfaces JO - Journal de l’École polytechnique — Mathématiques PY - 2022 SP - 581 EP - 600 VL - 9 PB - Ecole polytechnique UR - http://www.numdam.org/articles/10.5802/jep.190/ DO - 10.5802/jep.190 LA - en ID - JEP_2022__9__581_0 ER -
%0 Journal Article %A El Emam, Christian %A Seppi, Andrea %T Rigidity of minimal Lagrangian diffeomorphisms between spherical cone surfaces %J Journal de l’École polytechnique — Mathématiques %D 2022 %P 581-600 %V 9 %I Ecole polytechnique %U http://www.numdam.org/articles/10.5802/jep.190/ %R 10.5802/jep.190 %G en %F JEP_2022__9__581_0
El Emam, Christian; Seppi, Andrea. 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. http://www.numdam.org/articles/10.5802/jep.190/
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