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On Positively Curved Einstein Four-Manifolds with Euler Characteristic at Most Three

  • Liang Cheng
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Abstract

It is conjectured that a complete, orientable Einstein four-manifold with strictly positive sectional curvature must be isometric to either the round four-sphere S4 or the complex projective plane ℂℙ2 with the Fubini--Study metric. In this paper, we confirm this conjecture when the manifold is homeomorphic to S4 or ℂℙ2. More precisely, we show that a complete, orientable Einstein four-manifold M with strictly positive sectional curvature and Euler characteristic χ(M) ≤ 3 (equivalently, by Freedman's classification theorem, homeomorphic to S4 or ℂℙ2) is isometric to either the round S4 or the Fubini--Study ℂℙ2. As an application, we obtain that a complete, orientable Einstein four-manifold with strictly positive sectional curvature admitting a nontrivial Killing field is isometric to either the round S4 or the Fubini--Study ℂℙ2.

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Cite

BibTeXRIS
BibTeX
@misc{cheng2026positively,
  title = {{On Positively Curved Einstein Four-Manifolds with Euler Characteristic at Most Three}},
  author = {Liang Cheng},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23337},
  eprint = {2609.23337},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23337},
}
RIS
TY  - UNPB
TI  - On Positively Curved Einstein Four-Manifolds with Euler Characteristic at Most Three
AU  - Liang Cheng
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.23337
UR  - https://arxiv.org/abs/2609.23337
AB  - It is conjectured that a complete, orientable Einstein four-manifold with strictly positive sectional curvature must be isometric to either the round four-sphere S^4 or the complex projective plane ℂℙ^2 with the Fubini--Study metric. In this paper, we confirm this conjecture when the manifold is homeomorphic to S^4 or ℂℙ^2. More precisely, we show that a complete, orientable Einstein four-manifold M with strictly positive sectional curvature and Euler characteristic χ(M) ≤ 3 (equivalently, by Freedman's classification theorem, homeomorphic to S^4 or ℂℙ^2) is isometric to either the round S^4 or the Fubini--Study ℂℙ^2. As an application, we obtain that a complete, orientable Einstein four-manifold with strictly positive sectional curvature admitting a nontrivial Killing field is isometric to either the round S^4 or the Fubini--Study ℂℙ^2.
ER  -