Abstract
The energy of a Kähler class, on a compact complex manifold (M,J) of Kähler type, is the infimum of the squared L2-norm of the scalar curvature over all Kähler metrics representing the class. We study general properties of this functional, and define its gradient flow over all Kähler classes represented by metrics of fixed volume. When besides the trivial holomorphic vector field of (M,J), all others have no zeroes, we extend it to a flow over all cohomology classes of fixed top cup product. We prove that the dynamical system in this space defined by the said flow does not have periodic orbits, that its only fixed points are critical classes of a suitably defined extension of the energy function, and that along solution curves in the Kähler cone the energy is a monotone function. If the Kähler cone is forward invariant under the flow, solutions to the flow equation converge to a critical point of the class energy function. We show that this is always the case when the manifold has a signed first Chern class. We characterize the forward stability of the Kähler cone in terms of the value of a suitable time dependent form over irreducible subvarieties of (M,J). We use this result to draw several geometric conclusions, including the determination of optimal dimension dependent bounds for the squared L2-norm of the scalar curvature functional.
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References
Aubin, T.: Equations du type Monge-Ampère sur les varietès Kähleriennes compactes. C.R.A.S. Paris 283A, 119 (1976)
Bando, S., Mabuchi, T.: Uniqueness of Einstein Kähler metrics modulo connected group actions. Algebraic Geometry, Sendai, 1985, Adv. Stud. Pure Math. 10, Amsterdam: North-Holland, 1987, pp. 11–40
Barth, W., Peters, C., van de Ven, A.: Compact Complex Surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete 4, Berlin-Heidelberg-New York: Springer Verlag, 1984
Buchdahl, N.: On compact Kähler surfaces. Ann. Inst. Fourier, Grenoble 49, 287–302 (1999)
Calabi, E.: Extremal Kähler Metrics. In: S.-T. Yau (ed.), Seminar on Differential Geometry, Princeton, 1982
Demailly, J.P., Paun, M.: Numerical characterization of the Kähler cone of a compact Kähler manifold. http://arxiv.org/abs/math.AG/0105176, 2001
Derdziński, A.: Self-dual Kähler manifolds and Einstein manifolds of dimension four. Comp. Math. 49, 405–433 (1983)
Futaki, A.: Kähler-Einstein metrics and integral invariants. Lect. Notes in Math. 1314, Berlin-Heidelberg-New York: Springer-Verlag, 1987
Futaki, A., Mabuchi, T.: Bilinear forms and extremal Kähler vector fields associated with Kähler classes. Math. Annalen 301, 199–210 (1995)
Griffiths, P., Harris, J.: Principles of Algebraic Geometry. New York: Wiley-Interscience, 1978
Hitchin, N.: On compact four-dimensional Einstein manifolds. J. Diff. Geom. 9, 435–442 (1974)
Lamari, A.: Le cône kählérien d’une surface. J. Math. Pures Appl. 78, 249–263 (1999)
LeBrun, C.: 4-Manifolds without Einstein Metrics. Math. Res. Lett. 3, 133–147 (1996)
Petean, J.: Computations of the Yamabe invariant. Math. Res. Lett. 5, 703–709 (1998)
Simanca, S.R.: Precompactness of the Calabi Energy. Internat. J. Math. 7, 245–254 (1996)
Simanca, S.R.: Strongly extremal Kähler metrics. Ann. Global Anal. Geom. 18, 29–46 (2000)
Simanca, S.R., Stelling, L.: Canonical Kähler classes. Asian J. Math. 5, 585–598 (2001)
Tian, G.: Kähler-Einstein metrics with positive scalar curvature. Invent. Math. 130, 1–37 (1997)
Tian, G.: On Calabi’s conjecture for complex surfaces with positive first Chern class. Invent. Math. 101, 101–172 (1990)
Yau, S.T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I. Commun. Pure. Applied Math. 31, 339–411 (1978)
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Communicated by L. Takhtajan
Acknowledgement We would like to thank Nicholas Buchdahl for helpful conversations leading us to several improvements of an earlier version of the article, including the correction of two improper assertions.
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Simanca, S., Stelling, L. The Dynamics of the Energy of a Kähler Class. Commun. Math. Phys. 255, 363–389 (2005). https://doi.org/10.1007/s00220-004-1276-3
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DOI: https://doi.org/10.1007/s00220-004-1276-3