Obstacles for Sobolev-homeomorphisms with low rank -- pointwise a.e. vs distributional Jacobians

We show that for any k and s > k+1/k+2 there exist neither Ws,k/s-Sobolev nor Cˢ-H\"older homeomorphisms from the disk Bⁿ into RN whose gradient has rank $< k$ in distributional sense. This complements known examples of such kind of homeomorphisms whose gradient has rank $<k$ almost everywhere.