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The curl-Yamabe problem

  • Guofang Wang
  • Mingwei Zhang
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Abstract

We introduce and study a Yamabe-type variational problem associated with the curl operator on middle-degree forms on a closed oriented Riemannian manifold (Mn,g) with n≡3pmod4. The corresponding curl--Yamabe constant is defined by minimizing a conformally invariant, gauge-invariant quotient built from |curl α|2n/(n+1) and ∫M⟨curl α,α⟩, and its Euler--Lagrange equation is the critical curl--Yamabe equation curl α=(n+1)/2|α|2/(n−1)α. Using sharp curl--Sobolev inequalities on the sphere and an Aubin-type cut-off construction, we prove that 0<Ycurl(M,[g])≤ Ycurl(𝕊n) for every such manifold, and that the infimum is attained whenever Ycurl(M,[g])<Ycurl(𝕊n). Moreover, for n>3 we establish the strict inequality for manifolds that are not locally conformally flat, via a test-form expansion detecting the Weyl tensor, and deduce existence of a minimizer solving the curl--Yamabe equation.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026curlyamabe,
  title = {{The curl-Yamabe problem}},
  author = {Guofang Wang and Mingwei Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.13998},
  eprint = {2609.13998},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.13998},
}
RIS
TY  - UNPB
TI  - The curl-Yamabe problem
AU  - Guofang Wang
AU  - Mingwei Zhang
PY  - 2026
DA  - 2026/09/12
PB  - arXiv
DO  - 10.48550/arxiv.2609.13998
UR  - https://arxiv.org/abs/2609.13998
AB  - We introduce and study a Yamabe-type variational problem associated with the curl operator on middle-degree forms on a closed oriented Riemannian manifold (M^n,g) with n≡3pmod4. The corresponding curl--Yamabe constant is defined by minimizing a conformally invariant, gauge-invariant quotient built from |curl α|_2n/(n+1) and ∫_M⟨curl α,α⟩, and its Euler--Lagrange equation is the critical curl--Yamabe equation curl α=(n+1)/2|α|^2/(n−1)α. Using sharp curl--Sobolev inequalities on the sphere and an Aubin-type cut-off construction, we prove that 0<Y_curl(M,[g])≤ Y_curl(𝕊^n) for every such manifold, and that the infimum is attained whenever Y_curl(M,[g])<Y_curl(𝕊^n). Moreover, for n>3 we establish the strict inequality for manifolds that are not locally conformally flat, via a test-form expansion detecting the Weyl tensor, and deduce existence of a minimizer solving the curl--Yamabe equation.
ER  -