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Rigidity, sharp inequalities, and stability for σ2-curvature by a reference curvature technique

  • Wangzhe Wu
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Abstract

Using a reference-curvature technique, we prove four main theorems. (i) For n≥4, we remove the mean-curvature pinching from the Case-Wang rigidity theorem for σ2-flat conformal hemispheres with Ag∈Γ2+̅ and positive constant natural boundary curvature. (ii) On every closed positive Einstein manifold of dimension n≥5, each conformal metric of constant σ2-curvature is Einstein, and is homothetic to the background unless the background is a round sphere, thereby resolving the σ2-Obata problem posed by Chang, Gursky, and Yang. (iii) We extend the spherical constant-σ2/σ1 classification and sharp integral comparison to closed positive Einstein manifolds. The comparison holds throughout the positive-scalar-curvature conformal class, with equality only for homotheties on nonround backgrounds. (iv) We establish global quantitative σ2 stability on every fixed nonround positive Einstein background of dimension n≥5, throughout its positive-scalar-curvature conformal class.

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Cite

BibTeXRIS
BibTeX
@misc{wu2026rigidity,
  title = {{Rigidity, sharp inequalities, and stability for $σ_2$-curvature by a reference curvature technique}},
  author = {Wangzhe Wu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.10523},
  eprint = {2609.10523},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10523},
}
RIS
TY  - UNPB
TI  - Rigidity, sharp inequalities, and stability for σ_2-curvature by a reference curvature technique
AU  - Wangzhe Wu
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.10523
UR  - https://arxiv.org/abs/2609.10523
AB  - Using a reference-curvature technique, we prove four main theorems. (i) For n≥4, we remove the mean-curvature pinching from the Case-Wang rigidity theorem for σ_2-flat conformal hemispheres with A_g∈Γ_2^+̅ and positive constant natural boundary curvature. (ii) On every closed positive Einstein manifold of dimension n≥5, each conformal metric of constant σ_2-curvature is Einstein, and is homothetic to the background unless the background is a round sphere, thereby resolving the σ_2-Obata problem posed by Chang, Gursky, and Yang. (iii) We extend the spherical constant-σ_2/σ_1 classification and sharp integral comparison to closed positive Einstein manifolds. The comparison holds throughout the positive-scalar-curvature conformal class, with equality only for homotheties on nonround backgrounds. (iv) We establish global quantitative σ_2 stability on every fixed nonround positive Einstein background of dimension n≥5, throughout its positive-scalar-curvature conformal class.
ER  -