Singular extension of critical Sobolev mappings with values into complete Riemannian manifolds

Triggered by a recent criterion, due to A.~Petrunin [17], to check if a complete, non-compact, Riemannian manifold admits an isometric embedding into a Euclidean space with positive reach, we extend to manifolds with such property the singular extension results of B.~Bulanyi and J.~Van~Schaftingen [5] for maps in the critical, nonlinear Sobolev space \(W^{m/(m+1),m+1}\left(X^m,\mathcal{N}\right)\), where \(m \in \mathbb{N} \setminus \{0\}\), \(\mathcal{N}\) is a compact Riemannian manifold, and \(X^m\) is either the sphere \(\mathbb{S}^m = \partial \mathbb{B}^{m+1}_+\), the plane \(\mathbb{R}^m\), or again \(\mathbb{S}^m\) but seen as the boundary sphere of the Poincar\'{e} ball model of the hyperbolic space \(\mathbb{H}^{m+1}\). As in [5], we obtain that the extended maps satisfy an exponential weak-type Sobolev-Marcinkiewicz estimate. Finally, we provide some illustrative examples.