The generalized Lichnerowicz problem: Uniformly quasiregular mappings and space forms

A uqr mapping of an $n$-manifold $M$ is a mapping which is rational with respect to a bounded measurable conformal structure on $M$. Remarkably, the only closed manifolds on which locally (but not globally) injective uqr mappings act are Euclidean space forms. We further characterize space forms admitting uniformly quasiregular self mappings and we show that the space forms admitting branched uqr maps are precisely the spherical space forms. We further show that every non-injective uqr map of a Euclidean space form is a quasiconformal conjugate of a conformal map. This is not true if the non-injective hypothesis is removed.