Let ω be a connected and simply connected open subset of endowed with a Riemannian metric. Under a smoothness assumption on the boundary of ω, we first establish the existence and uniqueness up to isometries of an isometric immersion of ω into the Euclidean space , ‘up to the boundary’ of ω. When ω is bounded, we also show that the mapping that associates with the prescribed geometrical data the reconstructed submanifold is locally Lipschitz-continuous with respect to the topology of the Banach spaces .
Soit ω un ouvert connexe et simplement connexe de , muni d'une métrique riemannienne. Sous une certaine hypothèse de régularité sur la frontière de ω, on établit d'abord l'existence et l'unicité aux isométries près d'une immersion isométrique de ω dans l'espace euclidien , « jusqu'au bord » de ω. Lorsque ω est borné, on montre aussi que l'application qui associe aux données géométriques prescrites la sous-variété ainsi reconstruite est localement lipschitzienne pour les topologies usuelles des espaces de Banach .
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Marcela Szopos 1
@article{CRMATH_2004__339_4_265_0, author = {Marcela Szopos}, title = {On the recovery and continuity of a submanifold with boundary in higher dimensions}, journal = {Comptes Rendus. Math\'ematique}, pages = {265--270}, publisher = {Elsevier}, volume = {339}, number = {4}, year = {2004}, doi = {10.1016/j.crma.2004.05.022}, language = {en}, }
Marcela Szopos. On the recovery and continuity of a submanifold with boundary in higher dimensions. Comptes Rendus. Mathématique, Volume 339 (2004) no. 4, pp. 265-270. doi : 10.1016/j.crma.2004.05.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.05.022/
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