The extension of traces for Sobolev mappings between manifolds

The compact Riemannian manifolds M and N for which the trace operator from the first-order Sobolev space of mappings Ẇ1, p (M, N) to the fractional Sobolev-Slobodecki{} space Ẇ1 - 1/p, p (∂ M, N) is surjective when $1 < p < m$ are characterised. The traces are extended using a new construction which can be carried out assuming the absence of the known topological and analytical obstructions. When p ≥ m the same construction provides a Sobolev extension with linear estimates for maps that have a continuous extension, provided that there are no known analytical obstructions to such a control.