Cohomogeneity one conformal actions on the $3$-sphere and the three-dimensional Euclidean space

The aim of this paper is to classify cohomogeneity one (and zero) conformal actions on the three-dimensional Riemannian spaces whose conformal groups are essential, up to orbit equivalence. Among other results, for any such Riemannian space $M$, the representations of connected Lie groups acting with cohomogeneity one or zero on $M$ are determined in the conformal group of $M$ up to conjugacy and then up to orbit equivalence. Also the orbits are specified up to conformal equivalence.