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PreprintarXiv licencearXiv

Axially Symmetric Rigidity for a Q-Curvature-Type Equation on $ \mathbb{S}^N $

  • Changfeng Gui
  • Tuoxin Li
  • Juncheng Wei
  • Zikai Ye
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Abstract

We prove that for every even integer N≥ 4 or N∈{3,5,7}, every axially symmetric solution to the Q-curvature-type problem αPN u + (N−1)!(1−eNu/(∫𝕊N eNudw))=0 on 𝕊N is constant, provided that $ α\ge\frac{1}{2}$ and αnot =1. The proof starts from a Gegenbauer expansion and its weighted ℓ 2 estimate on coefficients. Then one key estimate is the refined estimate of the seminorm ⌊ G⌋2 using a specially designed integration-by-parts identity for general even integer N≥ 4. For N∈{3,5,7}, the seminorm estimates are established via a descent to a three--dimensional nonlocal difference quotient identity.

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Cite

BibTeXRIS
BibTeX
@misc{gui2026axially,
  title = {{Axially Symmetric Rigidity for a $Q$-Curvature-Type Equation on $ \mathbbS^N $}},
  author = {Changfeng Gui and Tuoxin Li and Juncheng Wei and Zikai Ye},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26227},
  eprint = {2609.26227},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26227},
}
RIS
TY  - UNPB
TI  - Axially Symmetric Rigidity for a Q-Curvature-Type Equation on $ \mathbb{S}^N $
AU  - Changfeng Gui
AU  - Tuoxin Li
AU  - Juncheng Wei
AU  - Zikai Ye
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.26227
UR  - https://arxiv.org/abs/2609.26227
AB  - We prove that for every even integer N≥ 4 or N∈{3,5,7}, every axially symmetric solution to the Q-curvature-type problem αP_N u + (N−1)!(1−e^Nu/(∫_𝕊^N e^Nudw))=0 on 𝕊^N is constant, provided that $ α\ge\frac{1}{2}$ and αnot =1. The proof starts from a Gegenbauer expansion and its weighted ℓ ^2 estimate on coefficients. Then one key estimate is the refined estimate of the seminorm ⌊ G⌋^2 using a specially designed integration-by-parts identity for general even integer N≥ 4. For N∈{3,5,7}, the seminorm estimates are established via a descent to a three--dimensional nonlocal difference quotient identity.
ER  -