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On the recovery and continuity of a submanifold with boundary in higher dimensions. (English. Abridged French version) Zbl 1064.53037

Let \(\Omega\) be a connected and simply connected open subset of \(\mathbb R^p\) endowed with a Riemannian metric. It is a classical result in differential geometry that \(\Omega\) can be immersed in Euclidean space \(\mathbb R^{p+q}\) if and only if the associated tensors satisfy the equations of Gauss, Ricci and Codazzi. Recently, it has been studied under which condition the reconstruction of a submanifold of \(\mathbb R^{p+q}\) can be done “up to the boundary” [see e.g. P. G. Ciarlet and C. Mardare, C. R., Math., Acad. Sci. Paris 338, No. 4, 333–340 (2004; Zbl 1057.53013)].
In the present paper, the existence and uniqueness up to isometries of an isometric immersion of \(\Omega\) into the Euclidean space \(\mathbb R^{p+q}\), “up to the boundary” of \(\Omega\) are established, under a smoothness assumption on the boundary of \(\Omega\). Moreover, if \(\Omega\) is bounded, it is shown that the mapping that associates the reconstructed submanifold with the prescribed geometrical data is locally Lipschitz-continuous with respect to the topology of the Banach spaces \(C^\ell (\overline\Omega)\), \(\ell\geq 1\).

MSC:

53C40 Global submanifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Citations:

Zbl 1057.53013
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References:

[1] Betounes, D. E., Differential geometric aspects of continuum mechanics in higher dimensions, (Differential Geometry: The Interface between Pure and Applied Mathematics. Differential Geometry: The Interface between Pure and Applied Mathematics, Contemp. Math., vol. 68 (1987)), 23-37
[2] do Carmo, M., Riemannian Geometry (1992), Birkhäuser: Birkhäuser Boston · Zbl 0752.53001
[3] Ciarlet, Ph. G.; Mardare, C., Recovery of a manifold with boundary and its continuity as a function of its metric tensor, C. R. Acad. Sci. Paris, Ser. I, 338, 333-340 (2004) · Zbl 1057.53013
[4] Ph.G. Ciarlet, C. Mardare, A surface as a function of its two fundamental forms, in press; Ph.G. Ciarlet, C. Mardare, A surface as a function of its two fundamental forms, in press · Zbl 1083.53007
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