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 Wm/(m+1),m+1(Xᵐ,N), where m ∈ N \0\, N is a compact Riemannian manifold, and Xᵐ is either the sphere Sᵐ = ∂ Bᵐ⁺¹₊, the plane Rᵐ, or again Sᵐ but seen as the boundary sphere of the Poincar\'{e} ball model of the hyperbolic space Hᵐ⁺¹. As in [5], we obtain that the extended maps satisfy an exponential weak-type Sobolev-Marcinkiewicz estimate. Finally, we provide some illustrative examples.