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Obstacles for Sobolev-homeomorphisms with low rank

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