Einstein Hermitian metrics of positive sectional curvature

Koca, Caner

doi:10.1090/S0002-9939-2014-11929-0

Einstein Hermitian metrics of positive sectional curvature

Author: Caner Koca
Journal: Proc. Amer. Math. Soc. 142 (2014), 2119-2122
MSC (2010): Primary 53C25, 53C55
DOI: https://doi.org/10.1090/S0002-9939-2014-11929-0
Published electronically: March 11, 2014
MathSciNet review: 3182029
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that, up to scaling and isometry, the only complete 4-manifold with an Einstein metric of positive sectional curvature which is also Hermitian with respect to some complex structure is the complex projective plane $\mathbb{CP}_2$ , equipped with its Fubini-Study metric.

[1] Aldo Andreotti, On the complex structures of a class of simply-connected manifolds, Algebraic geometry and topology. A symposium in honor of S. Lefschetz, Princeton University Press, Princeton, N. J., 1957, pp. 53–77. MR 0086357
[2] Thierry Aubin, Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252, Springer-Verlag, New York, 1982. MR 681859
[3] Lionel Bérard-Bergery, Sur de nouvelles variétés riemanniennes d’Einstein, Institut Élie Cartan, 6, Inst. Élie Cartan, vol. 6, Univ. Nancy, Nancy, 1982, pp. 1–60 (French). MR 727843
[4] Marcel Berger, Les variétés kählériennes compactes d’Einstein de dimension quatre à courbure positive, Tensor (N.S.) 13 (1963), 71–74 (French). MR 0155274
[5] Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684
[6] Eugenio Calabi, Extremal Kähler metrics. II, Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 95–114. MR 780039
[7] Xiuxiong Chen, Claude Lebrun, and Brian Weber, On conformally Kähler, Einstein manifolds, J. Amer. Math. Soc. 21 (2008), no. 4, 1137–1168. MR 2425183, https://doi.org/10.1090/S0894-0347-08-00594-8
[8] Theodore Frankel, Manifolds with positive curvature, Pacific J. Math. 11 (1961), 165–174. MR 0123272
[9] Victor Guillemin, Moment maps and combinatorial invariants of Hamiltonian 𝑇ⁿ-spaces, Progress in Mathematics, vol. 122, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1301331
[10] Matthew J. Gursky and Claude Lebrun, On Einstein manifolds of positive sectional curvature, Ann. Global Anal. Geom. 17 (1999), no. 4, 315–328. MR 1705915, https://doi.org/10.1023/A:1006597912184
[11] Shoshichi Kobayashi, Fixed points of isometries, Nagoya Math. J. 13 (1958), 63–68. MR 0103508
[12] Claude LeBrun, On Einstein, Hermitian 4-manifolds, J. Differential Geom. 90 (2012), no. 2, 277–302. MR 2899877
[13] Claude LeBrun, Einstein metrics on complex surfaces, Geometry and physics (Aarhus, 1995) Lecture Notes in Pure and Appl. Math., vol. 184, Dekker, New York, 1997, pp. 167–176. MR 1423163
[14] D. Page, A compact rotating gravitational instanton, Phys. Let. 79B (1978), no. 3, 235-238.
[15] Yum Tong Siu and Shing Tung Yau, Compact Kähler manifolds of positive bisectional curvature, Invent. Math. 59 (1980), no. 2, 189–204. MR 577360, https://doi.org/10.1007/BF01390043
[16] J. L. Synge, On the connectivity of spaces of positive curvature, Quart. J. Math. (Oxford series) 7 (1936), no. 1, 316-320.
[17] G. Tian, On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990), no. 1, 101–172. MR 1055713, https://doi.org/10.1007/BF01231499
[18] Burkhard Wilking, Manifolds with positive sectional curvature almost everywhere, Invent. Math. 148 (2002), no. 1, 117–141. MR 1892845, https://doi.org/10.1007/s002220100190
[19] Shing Tung Yau, Calabi’s conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 5, 1798–1799. MR 0451180
[20] 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, https://doi.org/10.1002/cpa.3160310304