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

Relying on a recent criterion, due to A.~Petrunin [13], to check if a complete, non-compact, Riemannian manifold admits an isometric embedding with positive reach into a Euclidean space, we extend to manifolds with such a property the singular extension results of B.~Bulanyi and J.~Van~Schaftingen [5] for maps in the critical, nonlinear Sobolev space $W^{m\mathrm{/}(m+1),m+1}(X^m,\mathcal{N})$, where $m\in \mathbb{N}\setminus \{0\}$, $\mathcal{N}$ is a compact Riemannian manifold, and $X^m$ is either the sphere ${\mathbb{S}}^m=\mathrm{\partial }{\mathbb{B}}_{+}^{m+1}$, the plane ${\mathbb{R}}^m$, or again ${\mathbb{S}}^m$ but seen as the boundary sphere of the Poincaré 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 of non-compact target manifolds to which our results apply.