Sobolev mixing constants and compactness in rational moduli space

We study uniformity of Sobolev mixing estimates for rational maps and uniformly quasiregular mappings. For rational maps of fixed degree, the optimal centered Sobolev trace constant $A_f$ is a continuous proper function on Möbius moduli space. Uniform bounds on $A_f$ therefore characterize relative compactness in moduli, and a minimizing class exists in every degree. In the setting of uniformly quasiregular endomorphisms of degree $d\ge 2$ on closed $n$-manifolds with an invariant conformal structure, the $k$th centered transfer operator from critical Sobolev energy into $L^1$ of the equilibrium measure has norm $A_f{d}^{-k\mathrm{/}n}$. On the mean-zero Sobolev space, the spectrum and Fredholm essential spectrum are the closed disk of radius $d^{-1\mathrm{/}n}$, with infinite-dimensional eigenspaces throughout its interior. The proofs use the energy scaling of pullback, a bounded equilibrium trace, and an obstruction from atoms of intermediate mass. Combined with conformal barycenter normalization and DeMarco–Faber’s degeneration theorem, this obstruction gives the moduli compactness criterion. Explicit families illustrate the distinction between degeneration and concentration caused by changes of coordinates.