On the recovery and continuity of a submanifold with boundary in higher dimensions (Q1887021)

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. \textit{P. G. Ciarlet} and \textit{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 \(C^\ell (\overline\Omega)\), \(\ell\geq 1\).