Regularity via minors and applications to conformal maps

We prove that if the minors of degree $k$ of a Sobolev map $\mathbb{R}^d \to \mathbb{R}^d$ are smooth then the map is smooth, when $k,d$ are not both even. We use this result to derive a simple, self-contained proof of the famous Liouville theorem for conformal maps, under the weakest possible regularity assumptions, in even dimensions which are not multiple of $4$. This is based on the approach taken by Iwaniec and Martin in [Acta Mathematica, 170(1):29--81, 1993]. We also prove the regularity of $W^{1,d/2}$ conformal maps between Riemannian manifolds, under the additional assumption of continuity.