Heat flows for extremal Kähler metrics
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 2, pp. 187-217.

Let (M,J,Ω) be a closed polarized complex manifold of Kähler type. Let G be the maximal compact subgroup of the automorphism group of (M,J). On the space of Kähler metrics that are invariant under G and represent the cohomology class Ω, we define a flow equation whose critical points are the extremal metrics, i.e. those that minimize the square of the L 2 -norm of the scalar curvature. We prove that the dynamical system in this space of metrics defined by the said flow does not have periodic orbits, and that its only fixed points are the extremal metrics. We prove local time-existence of the flow, and conclude that if the lifespan of the solution is finite, then the supremum of the norm of its curvature tensor must blow up as time approaches it. While doing this, we also prove that extremal solitons can only exist in the non-compact case, and that the range of the holomorphy potential of the scalar curvature is an interval independent of the metric chosen to represent Ø. We end up with some conjectures concerning the plausible existence and convergence of global solutions under suitable geometric conditions.

Classification : 53C55, 35K55, 58E11, 58J35
Simanca, Santiago R. 1

1 Institute for Mathematical Sciences Stony Brook, NY 11794
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     title = {Heat flows for extremal {K\"ahler} metrics},
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Simanca, Santiago R. Heat flows for extremal Kähler metrics. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 2, pp. 187-217. http://www.numdam.org/item/ASNSP_2005_5_4_2_187_0/

[1] M. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), 1-15. | MR | Zbl

[2] D. Burns and P. De Bartolomeis, Stability of vector bundles and extremal metrics, Invent. Math. 92 (1988), 403-407. | MR | Zbl

[3] E. Calabi, Extremal Kähler metrics, In: “Seminars on Differential Geometry”, S. T. Yau (ed.), Annals of Mathematics Studies, Princeton University Press, 1982, 259-290. | MR | Zbl

[4] E. Calabi, “Extremal Kähler Metrics II”, In: “Differential Geometry and Complex Analysis”, Chavel & Farkas (eds.), Springer-Verlag, 1985, 95-114. | MR | Zbl

[5] H.-D. Cao, Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds, Invent. Math. 81 (1985), 359-372. | MR | Zbl

[6] J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160. | MR | Zbl

[7] A. Futaki, “Kähler-Einstein metrics and integral invariants”, Lect. Notes in Math. 1314, Springer-Verlag, 1987. | MR | Zbl

[8] A. Futaki and T. Mabuchi, Bilinear forms and extremal Kähler vector fields associated with Kähler classes, Math. Ann. 301 (1995), 199-210. | MR | Zbl

[9] V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math. 67 (1982), 491-513. | MR | Zbl

[10] R. Hamilton, The Ricci flow on surfaces, Contemp. Math., 71 (1988), 237-262. | MR | Zbl

[11] R. Hamilton, Three manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), 255-306. | MR | Zbl

[12] T. Kato, “Abstract differential equations and non-linear mixed problems”, Lezioni Fermiane, Scuola Norm. Sup. Pisa Cl. Sci., 1985. | MR | Zbl

[13] C. Lebrun and S. R. Simanca, On Kähler Surfaces of Constant Positive Scalar Curvature, J. Geom. Anal. 5 (1995), 115-127. | MR | Zbl

[14] C. Lebrun and S. R. Simanca, Extremal Kähler Metrics and Complex Deformation Theory, Geom. Func. Anal. 4 (1994), 298-336. | MR | Zbl

[15] C. Lebrun and S. R. Simanca, On the Kähler Classes of Extremal Metrics, In: “Geometry and Global Analysis”, First MSJ Intern. Res. Inst. Sendai, Japan, Kotake, Nishikawa and Schoen (eds.), 1993. | Zbl

[16] M. Levine, A remark on extremal Kähler metrics, J. Differential Geom. 21 (1986), 73-77. | MR | Zbl

[17] S. R. Simanca, A K-energy characterization of extremal Kähler metrics, Proc. Amer. Math. Soc. 128 (2000), 1531-1535. | MR | Zbl

[18] S. R. Simanca, Strongly extremal Kähler metrics, Ann. Global Anal. Geom. 18 (2000), 29-46. | MR | Zbl

[19] S. R. Simanca, Precompactness of the Calabi Energy, Internat. J. Math. 7 (1996), 245-254. | MR | Zbl

[20] S. R. Simanca and L. Stelling, Canonical Kähler classes, Asian J. Math. 5 (2001), 585-598. | MR | Zbl

[21] S. R. Simanca and L. Stelling, The dynamics of the energy of a Kähler class, Commun. Math. Phys. 255 (2005), 363-389. | MR | Zbl

[22] G. Tian, Kähler-Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997), 1-37. | MR | Zbl

[23] S. T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I, Comm. Pure. Applied Math. 31 (1978), 339-411. | MR | Zbl