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On optimal 4-dimensional metrics. (English) Zbl 1154.53027

The problem is studied, which simply connected oriented 4-manifolds admit anti-self dual Riemannian metrics of flat scalar curvature. The new key ingredient is a proof that the connected sum of five reverse-oriented projective planes \(\overline{\mathbb{C}\mathbb{P}_2}\) admits such metrics. It is proceeded on the basis of the main following theorem: A smooth compact simply connected 4-manifold admits the above said metrics if
\(*\) \(M\) is diffeomorphic to \(k\,\overline{\mathbb{C}\mathbb{P}_2}\), for some \(k\geq 5\).
\(*\) \(M\) is diffeomorphic to \(k\,\overline{\mathbb{C}\mathbb{P}_2}\# \overline{\mathbb{C}\mathbb{P}_2}\), for some \(k\geq 10\).
\(*\) \(M\) is diffeomorphic to a hyper-Kähler manifold K3.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
53C20 Global Riemannian geometry, including pinching
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[1] Ahlfors, L., Bers, L.: Riemann’s mapping theorem for variable metrics. Ann. Math. 72(2), 385–404 (1960) · Zbl 0104.29902 · doi:10.2307/1970141
[2] Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. R. Soc. Lond. Ser. A 362, 425–461 (1978) · Zbl 0389.53011 · doi:10.1098/rspa.1978.0143
[3] Barth, W., Peters, C., de Ven, A.V.: Compact Complex Surfaces. Springer, Berlin (1984) · Zbl 0718.14023
[4] Berger, M.: Riemannian Geometry During the Second Half of the Twentieth Century. American Mathematical Society, Providence (2000). Reprint of the 1998 original · Zbl 0928.53001
[5] Bers, L.: On boundaries of Teichmüller spaces and on Kleinian groups I. Ann. Math. 91(2), 570–600 (1970) · Zbl 0197.06001 · doi:10.2307/1970638
[6] Bishop, C.J., Jones, P.W.: Hausdorff dimension and Kleinian groups. Acta Math. 179, 1–39 (1997) · Zbl 0921.30032 · doi:10.1007/BF02392718
[7] Calderbank, D.M.J., Singer, M.A.: Einstein metrics and complex singularities. Invent. Math. 156, 405–443 (2004) · Zbl 1061.53026 · doi:10.1007/s00222-003-0344-1
[8] Chen, X.X., LeBrun, C., Weber, B.: On conformally Kähler, Einstein manifolds. e-print arXiv:0705.0710 (2007)
[9] Chuckrow, V.: On Schottky groups with applications to Kleinian groups. Ann. Math. 88(2), 47–61 (1968) · Zbl 0186.40603 · doi:10.2307/1970555
[10] Donaldson, S., Friedman, R.: Connected sums of self-dual manifolds and deformations of singular spaces. Nonlinearity 2, 197–239 (1989) · Zbl 0671.53029 · doi:10.1088/0951-7715/2/2/002
[11] Donaldson, S.K.: An application of gauge theory to four-dimensional topology. J. Differ. Geom. 18, 279–315 (1983) · Zbl 0507.57010
[12] Eastwood, M.G., Singer, M.A.: The Fröhlicher [Frölicher] spectral sequence on a twistor space. J. Differ. Geom. 38, 653–669 (1993) · Zbl 0783.53040
[13] Eguchi, T., Hanson, A.J.: Self-dual solutions to Euclidean gravity. Ann. Phys. 120, 82–106 (1979) · Zbl 0409.53020 · doi:10.1016/0003-4916(79)90282-3
[14] Floer, A.: Self-dual conformal structures on CP2. J. Differ. Geom. 33, 551–573 (1991) · Zbl 0736.53046
[15] Freedman, M.: On the topology of 4-manifolds. J. Differ. Geom. 17, 357–454 (1982) · Zbl 0528.57011
[16] Gehring, F.W., Marshall, T.H., Martin, G.J.: The spectrum of elliptic axial distances in Kleinian groups. Indiana Univ. Math. J. 47, 1–10 (1998) · Zbl 0914.30031 · doi:10.1512/iumj.1998.47.1433
[17] Gibbons, G.W., Hawking, S.W.: Classification of gravitational instanton symmetries. Commun. Math. Phys. 66, 291–310 (1979) · doi:10.1007/BF01197189
[18] Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley-Interscience, New York (1978) · Zbl 0408.14001
[19] Hitchin, N.J.: Polygons and gravitons. Math. Proc. Camb. Philos. Soc. 85, 465–476 (1979) · Zbl 0405.53016 · doi:10.1017/S0305004100055924
[20] Joyce, D.: Constant scalar curvature metrics on connected sums. Int. J. Math. Math. Sci. 2003, 405–450 (2003) · Zbl 1026.53019 · doi:10.1155/S016117120310806X
[21] Kalafat, M.: Scalar curvature and connected sums of self-dual 4-manifolds. e-print arXiv:math/0611769 (2006)
[22] Kim, J., LeBrun, C., Pontecorvo, M.: Scalar-flat Kähler surfaces of all genera. J. Reine Angew. Math. 486, 69–95 (1997) · Zbl 0876.53044
[23] Kodaira, K.: A theorem of completeness of characteristic systems for analytic families of compact submanifolds of complex manifolds. Ann. Math. 75(2), 146–162 (1962) · Zbl 0112.38404 · doi:10.2307/1970424
[24] Kovalev, A., Singer, M.: Gluing theorems for complete anti-self-dual spaces. Geom. Funct. Anal. 11, 1229–1281 (2001) · Zbl 1012.53026 · doi:10.1007/s00039-001-8230-8
[25] Kronheimer, P.B.: Instantons gravitationnels et singularités de Klein. C.R. Acad. Sci. Paris Sér. I Math. 303, 53–55 (1986) · Zbl 0591.53057
[26] Kuiper, N.H.: On conformally-flat spaces in the large. Ann. Math. 50(2), 916–924 (1949) · Zbl 0041.09303 · doi:10.2307/1969587
[27] Lafontaine, J.: Remarques sur les variétés conformément plates. Math. Ann. 259, 313–319 (1982) · Zbl 0478.53034 · doi:10.1007/BF01456943
[28] LeBrun, C.: On the topology of self-dual 4-manifolds. Proc. Am. Math. Soc. 98, 637–640 (1986) · Zbl 0606.53029
[29] LeBrun, C.: Counter-examples to the generalized positive action conjecture. Commun. Math. Phys. 118, 591–596 (1988) · Zbl 0659.53050 · doi:10.1007/BF01221110
[30] LeBrun, C.: Explicit self-dual metrics on \(\mathbb{C}\)\(\mathbb{P}\)2#\(\mathbb{C}\)\(\mathbb{P}\)2. J. Differ. Geom. 34, 223–253 (1991) · Zbl 0725.53067
[31] LeBrun, C.: Curvature functionals, optimal metrics, and the differential topology of four-manifolds. In: Different Faces of Geometry. Int. Math. Ser. (N.Y.), pp. 199–256. Kluwer/Plenum, New York (2004) · Zbl 1088.53024
[32] LeBrun, C., Simanca, S.R.: Extremal Kähler metrics and complex deformation theory. Geom. Funct. Anal. 4, 298–336 (1994) · Zbl 0801.53050 · doi:10.1007/BF01896244
[33] LeBrun, C., Singer, M.: Existence and deformation theory for scalar-flat Kähler metrics on compact complex surfaces. Invent. Math. 112, 273–313 (1993) · Zbl 0793.53067 · doi:10.1007/BF01232436
[34] LeBrun, C., Singer, M.: A Kummer-type construction of self-dual 4-manifolds. Math. Ann. 300, 165–180 (1994) · Zbl 0818.53034 · doi:10.1007/BF01450482
[35] Maskit, B.: On boundaries of Teichmüller spaces and on Kleinian groups II. Ann. Math. 91(2), 607–639 (1970) · Zbl 0197.06003 · doi:10.2307/1970640
[36] Maskit, B.: Isomorphisms of function groups. J. Anal. Math. 32, 63–82 (1977) · Zbl 0392.30028 · doi:10.1007/BF02803575
[37] Maskit, B.: Kleinian Groups. Springer, Berlin (1988) · Zbl 0627.30039
[38] Nayatani, S.: Patterson-Sullivan measure and conformally flat metrics. Math. Z. 225, 115–131 (1997) · Zbl 0868.53024 · doi:10.1007/PL00004301
[39] Penrose, R.: Nonlinear gravitons and curved twistor theory. Gen. Relativ. Gravit. 7, 31–52 (1976) · Zbl 0354.53025 · doi:10.1007/BF00762011
[40] Pontecorvo, M.: On twistor spaces of anti-self-dual Hermitian surfaces. Trans. Am. Math. Soc. 331, 653–661 (1992) · Zbl 0754.53053 · doi:10.2307/2154133
[41] Rollin, Y., Singer, M.: Non-minimal scalar-flat Kähler surfaces and parabolic stability. Invent. Math. 162, 235–270 (2005) · Zbl 1083.32021 · doi:10.1007/s00222-004-0436-6
[42] Schoen, R., Yau, S.-T.: Conformally flat manifolds, Kleinian groups and scalar curvature. Invent. Math. 92, 47–71 (1988) · Zbl 0658.53038 · doi:10.1007/BF01393992
[43] Taubes, C.H.: The existence of anti-self-dual conformal structures. J. Differ. Geom. 36, 163–253 (1992) · Zbl 0822.53006
[44] Trudinger, N.: Remarks concerning the conformal deformation of metrics to constant scalar curvature. Ann. Sc. Norm. Sup. Pisa 22, 265–274 (1968) · Zbl 0159.23801
[45] Yau, S.T.: On the curvature of compact Hermitian manifolds. Invent. Math. 25, 213–239 (1974) · Zbl 0299.53039 · doi:10.1007/BF01389728
[46] Yau, S.T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I. Commun. Pure Appl. Math. 31, 339–411 (1978) · Zbl 0369.53059 · doi:10.1002/cpa.3160310304
[47] Zhou, J.: Connected sums of self-dual orbifolds. Ph.D. thesis, State University of New York at Stony Brook (1995)
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