A counterexample to the local existence of conformal normal metrics

Abstract

For every dimension $n\ge3$, we construct a smooth Riemannian metric whose entire Taylor series at a point agrees with the Euclidean metric and whose conformal class contains no smooth conformal normal metric near that point. For the metric used in the construction, parametrized by a smooth angular function, we prove that conformal normalization forces a dimension-dependent exponential decay rate for the spherical harmonic components of that function. This gives counterexamples whenever the angular coefficients decay too slowly. In particular, every nonanalytic angular function yields a counterexample. The proof combines a weighted cylinder estimate with an integral identity relating the angular spectral coefficients of the background volume defect to the nonlinear remainder. We also give a direct counterexample to the linear right inverse used in Cao's smooth existence argument and identify the failure of angular summability in that construction.