Abstract
We prove the existence of infinite-dimensional families of(non-Kähler) almost-Kähler metrics with constant scalar curvature oncertain compact manifolds. These are obtained by deformingconstant-scalar-curvature Kähler metrics on suitable compact complexmanifolds. We prove several other similar results concerning the scalarcurvature and/or the *-scalar curvature. We also discuss thescalar curvature functions of almost-Kähler metrics.
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Apostolov, V. and Dr?ghici, T.: Hermitian conformal classes and Kähler structures on 4-manifolds, Differential Geom. Appl. 11 (1999), 179-195.
Apostolov, V., Gauduchon, P. and Grantcharov, G.: Bihermitian structures on complex surfaces, Proc. London Math. Soc. (3) 79 (1999), 414-428.
Apostolov, V., Armstrong, J. and Dr?ghici, T.: Local rigidity of certain classes of almost-Kähler 4-manifolds, Preprint 1999, e-print math.DG/9911197.
Armstrong, J.: Almost-Kähler geometry, Ph.D. Thesis, Oxford, 1998.
Aubin, T.: Equations différentielles non linéaires et problème de Yamabe concernant la couboure scalaire, J. Math. Pures Appl. (9) 55 (1976), 269-296.
Bourguignon, J. P., Ebin, D. and Marsden, J. E.: Sur le noyau des opérateurs pseudodifférentiels à symbole surjectif et non injectif, C. R. Acad. Sci., Paris (I) 282 (1976), 867-870.
Barth, W., Peters, C. and Van de Ven, A.: Compact Complex Surfaces, Springer-Verlag, Berlin, 1984.
Besse, A. L.: Einstein Manifolds, Ergeb. Math. Grenzgeb. (3) 10, Springer-Verlag, Berlin, 1987.
Blair, D.: The 'Total Scalar Curvature' as a symplectic invariant, and related results, in: N. K. Stephanidis (ed.), Proc. 3rd Congress of Geometry (Thessaloniki, 1991), Aristotle Univ., Thessaloniki, 1992, pp. 79-83.
Cordero, L. A., Fernandez, M. and de Leon, M.: Examples of compact non-Kähler almost-Kähler manifolds, Proc. Amer. Math. Soc. 95(2) (1985), 280-286.
Fujiki, A. and Schumacher, G.: The moduli space of extremal compact Kähler manifolds and generalized Weil-Petersson metric, Publ. Res. Inst. Math. Sci. 26(1) (1990), 101-183.
Gompf, R.: A new construction of symplectic manifolds, Ann. of Math. (2) 142(3) (1995), 527-595.
Howard, A.: Holomorphic vector fields on algebraic manifolds, Amer. J.Math. 94 (1972), 1282-1290.
Jelonek, W.: Some simple examples of almost-Kähler non-Kähler structures, Math. Ann. 305 (1996), 639-649.
Kazdan, J. L. and Warner, F.W.: A direct approach to the determination of Gaussian and scalar curvature functions, Invent. Math. 28 (1975), 227-230.
Kazdan, J. L. and Warner, F. W.: Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures, Ann. of Math. (2) 101 (1975), 317-331.
Kim, J., LeBrun, C. and Pontecorvo, M.: Scalar flat Kähler surfaces of all genera, J. Reine Angew. Math. 486 (1997), 69-95.
LeBrun, C.: Ricci curvature, minimal volumes, and Seiberg-Witten theory, Invent. Math. 145(2) (2001), 279-316.
LeBrun, C. and Simanca, S. R.: Extremal Kähler metrics and complex deformation theory, Geom. Funct. Anal. 4 (1994), 179-200.
LeBrun, C. and Simanca, S. R.: On Kähler surfaces of constant positive scalar curvature, J. Geom. Anal. 5 (1995), 115-127.
LeBrun, C. and Singer, M.: Existence and deformation theory for scalar-flat Kähler metrics on compact complex surfaces, Invent. Math. 112 (1993), 273-313.
McDuff, D. and Salamon, D.: Introduction to Symplectic Topology, Oxford Univ. Press, Oxford, 1998.
Morrey, C. B.: Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin, 1966.
Ornea, L. and Dragomir, S.: Locally Conformal Kähler Geometry, Progr. in Math. 155, Birkhäuser, Boston, 1998.
Palais, R.: Foundations of global Non-Linear Analysis, Benjamin, New York, 1968.
Sekigawa, K.: Some compact Einstein almost-Kähler manifolds, J. Math. Soc. Japan 39(4) (1987), 677-684.
Siu, Y. T.: The existence of Kähler-Einstein metrics on manifolds with positive anticanonical line bundle and a suitable finite symmetry group, Ann. of Math. (2) 127(3) (1988), 585-627.
Sung, C.: Extremal almost-Kähler metrics and Seiberg-Witten theory, Preprint, Stony Brook, 2001.
Thurston, W.: Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55(2) (1976), 467-468.
Tian, G.: On Calabi's conjecture for complex surfaces with positive first Chern class, Invent. Math. 101(1) (1990), 101-172.
Yau, S. T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I, Commun. Pure Appl. Math. 31(3) (1978), 339-411.
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Kim, J., Sung, C. Deformations of Almost-Kähler Metrics with Constant Scalar Curvature on Compact Kähler Manifolds. Annals of Global Analysis and Geometry 22, 49–73 (2002). https://doi.org/10.1023/A:1016217414800
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DOI: https://doi.org/10.1023/A:1016217414800