On the topology of self-dual 4-manifolds

LeBrun, Claude

doi:10.1090/S0002-9939-1986-0861766-2

On the topology of self-dual $4$-manifolds
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by Claude LeBrun

Proc. Amer. Math. Soc. 98 (1986), 637-640

DOI: https://doi.org/10.1090/S0002-9939-1986-0861766-2

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Abstract:

Obstructions are obtained for the problem of finding a Riemannian metric with self-dual conformal curvature and nonnegative scalar curvature on a given smooth compact $4$-manifold. A list is given of those simply connected manifolds conceivably admitting such metrics.

References

Richard Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), no. 2, 479–495. MR 788292
Nigel Hitchin, Harmonic spinors, Advances in Math. 14 (1974), 1–55. MR 358873, DOI 10.1016/0001-8708(74)90021-8
M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), no. 1711, 425–461. MR 506229, DOI 10.1098/rspa.1978.0143
Shing Tung Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411. MR 480350, DOI 10.1002/cpa.3160310304
Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
Shing Tung Yau, On the curvature of compact Hermitian manifolds, Invent. Math. 25 (1974), 213–239. MR 382706, DOI 10.1007/BF01389728
Daniel S. Freed and Karen K. Uhlenbeck, Instantons and four-manifolds, 2nd ed., Mathematical Sciences Research Institute Publications, vol. 1, Springer-Verlag, New York, 1991. MR 1081321, DOI 10.1007/978-1-4613-9703-8