On the recovery and continuity of a submanifold with boundary in higher dimensions

Marcela Szopos

doi:10.1016/j.crma.2004.05.022

Differential Geometry

On the recovery and continuity of a submanifold with boundary in higher dimensions

Presented by: Philippe G. Ciarlet

Marcela Szopos ¹

¹ Laboratoire Jacques-Louis Lions, université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France

Comptes Rendus. Mathématique, Volume 339 (2004) no. 4, pp. 265-270

Abstract (Original language)
Résumé

Let ω be a connected and simply connected open subset of $R^{p}$ 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 $R^{p + q}$ , ‘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 $C^{l} (\bar{ω}), l ⩾ 1$ .

Soit ω un ouvert connexe et simplement connexe de $R^{p}$ , 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 $R^{p + q}$ , « 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 $C^{l} (\bar{ω}), l ⩾ 1$ .

Received: 2004-05-19
Accepted: 2004-05-26
Published online: 2004-07-24

DOI: 10.1016/j.crma.2004.05.022

Author's affiliations:

Marcela Szopos ¹

¹ Laboratoire Jacques-Louis Lions, université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France

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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

References
Cited by

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[2] M. do Carmo Riemannian Geometry, Birkhäuser, Boston, 1992

[3] Ph.G. Ciarlet; C. Mardare Recovery of a manifold with boundary and its continuity as a function of its metric tensor, C. R. Acad. Sci. Paris, Ser. I, Volume 338 (2004), pp. 333-340

[4] Ph.G. Ciarlet, C. Mardare, A surface as a function of its two fundamental forms, in press

[5] H. Jacobowitz The Gauss–Codazzi equations, Tensor (N.S.), Volume 39 (1982), pp. 15-22

[6] M. Szopos, On the recovery and continuity of a submanifold with boundary, in press

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