Comptes Rendus
Differential Geometry
On the recovery and continuity of a submanifold with boundary in higher dimensions
Comptes Rendus. Mathématique, Volume 339 (2004) no. 4, pp. 265-270.

Let ω be a connected and simply connected open subset of Rp 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 Rp+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 Cl(ω¯),l1.

Soit ω un ouvert connexe et simplement connexe de Rp, 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 Rp+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 Cl(ω¯),l1.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.05.022
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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.05.022/

[1] D.E. Betounes Differential geometric aspects of continuum mechanics in higher dimensions, Differential Geometry: The Interface between Pure and Applied Mathematics, Contemp. Math., vol. 68, 1987, pp. 23-37

[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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