PreprintarXiv licencearXiv
Bach-flat metrics and global regularity of conformal structures
Abstract
We study global regularity properties of conformal structures on compact manifolds with applications to conformally invariant variational problems. We find sharp conditions ensuring that a manifold equipped with a Riemannian metric of Sobolev class is conformally diffeomorphic to a smooth one. In particular, we prove that W2,q Bach-flat four-manifolds with q > 2 are smooth up to conformal diffeomorphisms. We also solve the Yamabe problem for W2,q metrics with q > 2 on three-manifolds under a mild conformal condition.
Related
- Rigidity of conformal metrics with constant sixth-order Q-curvature
- Harmonic-Einstein maps and critical metrics for eigenvalues of the conformal Laplacian
- Conformal boundary rigidity for simple Finsler metrics
- Complete conformal metrics of constant positive scalar curvature with prescribed isolated singularities
- A counterexample to the local existence of conformal normal metrics
BibTeX
@misc{avalos2026bachflat,
title = {{Bach-flat metrics and global regularity of conformal structures}},
author = {Rodrigo Avalos and Albachiara Cogo and Andoni Royo Abrego},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35257},
eprint = {2610.35257},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35257},
}
RIS
TY - UNPB
TI - Bach-flat metrics and global regularity of conformal structures
AU - Rodrigo Avalos
AU - Albachiara Cogo
AU - Andoni Royo Abrego
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35257
UR - https://arxiv.org/abs/2610.35257
AB - We study global regularity properties of conformal structures on compact manifolds with applications to conformally invariant variational problems. We find sharp conditions ensuring that a manifold equipped with a Riemannian metric of Sobolev class is conformally diffeomorphic to a smooth one. In particular, we prove that W^2,q Bach-flat four-manifolds with q > 2 are smooth up to conformal diffeomorphisms. We also solve the Yamabe problem for W^2,q metrics with q > 2 on three-manifolds under a mild conformal condition.
ER -