Quantitative characterization of traces of Sobolev maps

We give a quantitative characterization of traces on the boundary of Sobolev maps in (W)over dot(1, p)(M, N), where M and Ar are compact Riemannian manifolds, partial derivative M not equal theta: the Borel-measurable maps u: partial derivative M -> N that are the trace of a map U is an element of (W)over dot(1, p) (M, N) are characterized as the maps for which there exists an extension energy density w: partial derivative M -> [0, infinity] that controls the Sobolev energy of extensions from [p - 1]-dimensional subsets of partial derivative M to [p]-dimensional subsets of M.