A sharp threshold for mixed Q-curvature rigidity

Abstract

We resolve Case's question (Crelle's Journal, 2024) on the sharp lower threshold for Obata-type rigidity of I_a=Q+aσ₂(A) in positive Einstein conformal classes of dimension n≥4. Using a new reference-curvature identity, we prove that, on a closed connected manifold, every smooth metric in such a class with nonnegative scalar curvature and constant I_a is Einstein whenever a≥−4. The scalar-curvature sign assumption can be removed for −4≤ a≤−2(n−2)/(n−1). We also extend the quotient rigidity theorem of Ge--Wang--Wei to I_a=ΛR^θ for R>0, a≥−4, and θ≤1. Sharpness is established in every dimension by a shooting construction. For each sufficiently small η>0, we construct a smooth non-Einstein metric in the round conformal class on 𝕊ⁿ with positive scalar curvature and the same constant I_−4−η as the unit round metric. The construction smoothly matches a perturbed Schwarzschild neck near the equator with perturbed round caps at both poles.

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BibTeX RIS

BibTeX

@misc{wu2026sharp,
  title = {{A sharp threshold for mixed $Q$-curvature rigidity}},
  author = {Wangzhe Wu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.10527},
  eprint = {2609.10527},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10527},
}

RIS

TY  - UNPB
TI  - A sharp threshold for mixed Q-curvature rigidity
AU  - Wangzhe Wu
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.10527
UR  - https://arxiv.org/abs/2609.10527
AB  - We resolve Case's question (Crelle's Journal, 2024) on the sharp lower threshold for Obata-type rigidity of I_a=Q+aσ_2(A) in positive Einstein conformal classes of dimension n≥4. Using a new reference-curvature identity, we prove that, on a closed connected manifold, every smooth metric in such a class with nonnegative scalar curvature and constant I_a is Einstein whenever a≥−4. The scalar-curvature sign assumption can be removed for −4≤ a≤−2(n−2)/(n−1). We also extend the quotient rigidity theorem of Ge--Wang--Wei to I_a=ΛR^θ for R>0, a≥−4, and θ≤1. Sharpness is established in every dimension by a shooting construction. For each sufficiently small η>0, we construct a smooth non-Einstein metric in the round conformal class on 𝕊^n with positive scalar curvature and the same constant I_−4−η as the unit round metric. The construction smoothly matches a perturbed Schwarzschild neck near the equator with perturbed round caps at both poles.
ER  -

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