Abstract
Let (M, g) be a compact oriented four-dimensional Einstein manifold. If M has positive intersection form and g has non-negative sectional curvature, we show that, up to rescaling and isometry, (M, g) is ℂℙ2, with its standard Fubini–Study metric.
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References
Anderson, M.: The L 2 structure of moduli spaces of Einstein metrics on 4-manifolds, Geom. Funct. Anal. 2 (1992), 29–89.
Besse, A.: Einstein Manifolds, Springer-Verlag, Berlin, 1987.
Bieberbach, L.: Über die Bewegungsgruppen der Euklidischen Räume I, Math. Ann. 70 (1911), 297–336; II, 72 (1912), 400–412.
Bishop, R. L. and Crittenden, R. J.: Geometry of Manifolds, Academic Press, New York, 1964.
Bourgignon, J. P.: Les variétés de dimension 4 à signature non nulle dont la courbure est harmonique sont d'Einstein, Invent. Math. 63 (1981), 263–286.
Cheeger, J.: Finiteness theorems for Riemannian manifolds, Amer. J. Math. 92 (1970), 61–74.
Derdziński, A.: Self-dual Kähler manifolds and Einstein manifolds of dimension four, Comp. Math. 49 (1983), 405–433.
DeTurck, D. and Kazdan, J.: Some regularity theorems in Riemannian geometry, Ann. Sci. École Norm. Sup. 14 (1981), 249–260.
Freedman, M.: On the topology of 4-manifolds, J. Differential Geom. 17 (1982), 357–454.
Friedrich, T. and Kurke, H.: Compact four-dimensional self-dual Einstein manifolds with positive scalar curvature, Math. Nachr. 106 (1982), 271–299.
Gursky, M.: The Weyl functional, DeRham cohomology, and Kähler-Einstein metrics, Preprint, 1996.
Gursky, M.: Four-manifolds with δW + = 0 and Einstein constants on the sphere, Preprint, 1997.
Gursky, M. and LeBrun, C.: Yamabe invariants and Spinc structures, Geom. Funct. Anal. (to appear).
Hitchin, N. J.: On compact four-dimensional Einstein manifolds, J. Differential Geom. 9 (1974), 435–442.
Hitchin, N. J.: Kählerian twistor spaces, Proc. London Math. Soc. 43 (1981), 133–150.
Koiso, N.: Rigidity and stability of Einstein metrics. The case of compact symmetric spaces, Osaka J. Math. 17 (1980), 51–73.
LeBrun, C.: Yamabe constants and the perturbed Seiberg-Witten equations, Comm. Anal. Geom. 5 (1997), 535–553.
Myers, S. B.: Riemannian manifolds with positive mean curvature, Duke Math. J. 8 (1941), 401–404.
Obata, M.: The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geom. 6 (1971), 247–258.
Penrose, R. and Rindler, W.: Spinors and Space-Time. Vol. 2: Spinor and Twistor Methods in Space-Time Geometry, Cambridge University Press, Cambridge, 1986.
Singer, I. M. and Thorpe, J. A.: The curvature of four-dimensional Einstein spaces, in Spencer, D. C. and Iyanaga, S. (eds), Global Analysis (Papers in Honor of K. Kodaira), University of Tokyo Press, Tokyo, 1969, pp. 355–365.
Synge, J. L.: On the connectivity of spaces of positive curvature, Quart. J. Math. 7 (1936), 316–320.
Thorpe, J. A.: Some remarks on the Gauss-Bonnet formula, J. Math. Mech. 18 (1969), 779–786.
Tian, G.: On Calabi's conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990), 101–172.
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Gursky, M.J., LeBrun, C. On Einstein Manifolds of Positive Sectional Curvature. Annals of Global Analysis and Geometry 17, 315–328 (1999). https://doi.org/10.1023/A:1006597912184
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DOI: https://doi.org/10.1023/A:1006597912184