A counterexample to the local existence of conformal normal metrics

Abstract

For every dimension n≥3, 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.

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BibTeX RIS

BibTeX

@misc{zhang2026counterexample,
  title = {{A counterexample to the local existence of conformal normal metrics}},
  author = {Zhe Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.24541},
  eprint = {2610.24541},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.24541},
}

RIS

TY  - UNPB
TI  - A counterexample to the local existence of conformal normal metrics
AU  - Zhe Zhang
PY  - 2026
DA  - 2026/10/17
PB  - arXiv
DO  - 10.48550/arxiv.2610.24541
UR  - https://arxiv.org/abs/2610.24541
AB  - For every dimension n≥3, 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.
ER  -

Abstract

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