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Critical metrics for the Dirac-Yamabe eigenvalue ratio and Einstein-Dirac equations

  • Mehdi Eddaoudi
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Abstract

Let (Mn,g) be a closed spin manifold of dimension n≥ 3 with positive Yamabe constant. We prove that critical metrics for the mixed eigenvalue ratio ℋ(g)=λ1(Dg)2/λ1(Lg), where Dg is the Dirac operator and Lg is the conformal Laplacian are characterized by a harmonic section whose energy-momentum tensor satisfies an Einstein--Dirac equation. We discuss consequences of this result for Hijazi's inequality, and we construct an example to prove that ℋ is unbounded above within the conformal class of the round sphere.

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Cite

BibTeXRIS
BibTeX
@misc{eddaoudi2026critical,
  title = {{Critical metrics for the Dirac-Yamabe eigenvalue ratio and Einstein-Dirac equations}},
  author = {Mehdi Eddaoudi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.33825},
  eprint = {2610.33825},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.33825},
}
RIS
TY  - UNPB
TI  - Critical metrics for the Dirac-Yamabe eigenvalue ratio and Einstein-Dirac equations
AU  - Mehdi Eddaoudi
PY  - 2026
DA  - 2026/10/24
PB  - arXiv
DO  - 10.48550/arxiv.2610.33825
UR  - https://arxiv.org/abs/2610.33825
AB  - Let (M^n,g) be a closed spin manifold of dimension n≥ 3 with positive Yamabe constant. We prove that critical metrics for the mixed eigenvalue ratio ℋ(g)=λ_1(D_g)^2/λ_1(L_g), where D_g is the Dirac operator and L_g is the conformal Laplacian are characterized by a harmonic section whose energy-momentum tensor satisfies an Einstein--Dirac equation. We discuss consequences of this result for Hijazi's inequality, and we construct an example to prove that ℋ is unbounded above within the conformal class of the round sphere.
ER  -