The sharp σk-curvature inequality on locally conformally flat manifolds in quantitative form
Abstract
Let 2≤ k<n/2 and let (Mn,[g]) be a smooth, closed, connected, and locally conformally flat Riemannian manifold with a k-admissible metric in the conformal class [g]. We prove a stability result of the σk-curvature inequality on M, in the sense that if equality is almost satisfied for some conformal metric, then its conformal factor is close to a minimizer of the inequality. Closeness is measured quantitatively in terms of Sobolev norms of the conformal factor, namely with respect to the W1,2- and the W1,2k-norm. In the non-degenerate case, these norms come with optimal exponents $2$ and 2k, respectively, whereas in general the exponents are 2+γ and max{2k,2+γ} for some γ≥ 0 originating from a Łojasiewicz inequality. This extends a previous result by Frank and the author from k=2 and the sphere to 2≤ k<n/2 and the full class of manifolds originally considered by Viaclovsky. It also extends a previous result by Engelstein--Neumayer--Spolaor from k=1 to the setting of fully non-linear scalar curvatures.
Related
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BibTeX
@misc{peteranderl2026sharp,
title = {{The sharp $σ_k$-curvature inequality on locally conformally flat manifolds in quantitative form}},
author = {Jonas W. Peteranderl},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13133},
eprint = {2609.13133},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13133},
}
RIS
TY - UNPB
TI - The sharp σ_k-curvature inequality on locally conformally flat manifolds in quantitative form
AU - Jonas W. Peteranderl
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.13133
UR - https://arxiv.org/abs/2609.13133
AB - Let 2≤ k<n/2 and let (M^n,[g]) be a smooth, closed, connected, and locally conformally flat Riemannian manifold with a k-admissible metric in the conformal class [g]. We prove a stability result of the σ_k-curvature inequality on M, in the sense that if equality is almost satisfied for some conformal metric, then its conformal factor is close to a minimizer of the inequality. Closeness is measured quantitatively in terms of Sobolev norms of the conformal factor, namely with respect to the W^1,2- and the W^1,2k-norm. In the non-degenerate case, these norms come with optimal exponents $2$ and 2k, respectively, whereas in general the exponents are 2+γ and max{2k,2+γ} for some γ≥ 0 originating from a Łojasiewicz inequality. This extends a previous result by Frank and the author from k=2 and the sphere to 2≤ k<n/2 and the full class of manifolds originally considered by Viaclovsky. It also extends a previous result by Engelstein--Neumayer--Spolaor from k=1 to the setting of fully non-linear scalar curvatures.
ER -