The Conformal Parametrization of Nonnegatively Curved Surfaces Is Not a Homeomorphism at Integer Smoothness

Let M_0 = ℂ and M_1 = S², with complete metrics g_κ of constant curvature κ ∈ {0, 1}. Every complete smooth metric of nonnegative curvature on M_κ can be written uniquely as φ*(e^{−2u} g_κ), where φ is an orientation-preserving diffeomorphism fixing 0 and 1 (and ∞ if κ = 1) and u is a smooth function. Work of Belegradek, Hu and Banakh shows that the map (u, φ) ↦ φ*(e^{−2u} g_κ) is a homeomorphism when metrics and functions carry the topology of C^{k+α} convergence on compact sets and diffeomorphisms that of C^{k+1+α} convergence, with 0 < α < 1. Belegradek asked whether this remains true for α = 0, and expected that it does not. We show that it fails for every integer k ≥ 0 and on both surfaces: the map is a continuous bijection whose inverse is discontinuous at every point, also on the subspace of positively curved metrics. For k = 0 this follows from an elementary spiral construction. For k ≥ 1 we use diffeomorphisms φ_ε = z + ε z^{k+1} H_ε(|z|²) whose Beltrami coefficient is approximately ε z^{k+1}/z̄, regularized at the scale e^{−1/ε}. They converge to the identity in C^k, while their (k+1)-st derivative at 0 tends to −2(k+1)!. The main work is to choose the conformal factors so that the metrics converge in C^k and keep nonnegative curvature; for k = 1 this needs an additional C¹-small conformal correction. The conformal factors then fail to converge in C^k as well. This is an unrefereed note.