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Sharp Yamabe Bounds for Spherical Three-Space Forms via Four-Dimensional Poincaré--Einstein Compactness

  • Jakob Loren
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Abstract

Let Γ<SO(4) be a nontrivial finite group acting freely on S3, and let M=S3/Γ. We prove that every smooth conformal class C on M satisfies Y(M,C)≤ |Γ|−2/3Y(S3), with equality for the quotient-round conformal class. Consequently, σ(M)=|Γ|−2/3Y(S3). Our approach uses four-dimensional Poincaré--Einstein fillings of the ball. We first establish compactness for fillings whose conformal infinities range over a compact family of positive-scalar-curvature metrics. Since compactness alone does not guarantee continuation through degenerate Einstein metrics, we use equivariant Fredholm parity to obtain a filling with the prescribed boundary conformal class and group action. A common interior fixed point then makes the Li--Qing--Shi boundary test function invariant, and its descent to M gives the sharp covering factor.

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Cite

BibTeXRIS
BibTeX
@misc{loren2026sharp,
  title = {{Sharp Yamabe Bounds for Spherical Three-Space Forms via Four-Dimensional Poincaré--Einstein Compactness}},
  author = {Jakob Loren},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18793},
  eprint = {2610.18793},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18793},
}
RIS
TY  - UNPB
TI  - Sharp Yamabe Bounds for Spherical Three-Space Forms via Four-Dimensional Poincaré--Einstein Compactness
AU  - Jakob Loren
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.18793
UR  - https://arxiv.org/abs/2610.18793
AB  - Let Γ<SO(4) be a nontrivial finite group acting freely on S^3, and let M=S^3/Γ. We prove that every smooth conformal class C on M satisfies Y(M,C)≤ |Γ|^−2/3Y(S^3), with equality for the quotient-round conformal class. Consequently, σ(M)=|Γ|^−2/3Y(S^3). Our approach uses four-dimensional Poincaré--Einstein fillings of the ball. We first establish compactness for fillings whose conformal infinities range over a compact family of positive-scalar-curvature metrics. Since compactness alone does not guarantee continuation through degenerate Einstein metrics, we use equivariant Fredholm parity to obtain a filling with the prescribed boundary conformal class and group action. A common interior fixed point then makes the Li--Qing--Shi boundary test function invariant, and its descent to M gives the sharp covering factor.
ER  -