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 (Xᵐ, N), where m N \0\, N is a compact Riemannian manifold, and Xᵐ is either the sphere Sᵐ = B^m+1_+, the plane Rᵐ, or again Sᵐ but seen as the boundary sphere of the Poincar\'e ball model of the hyperbolic space 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.