Stability of the Calabi flow near an extremal metric

Huang, Hongnian; Zheng, Kai

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 1, pp. 167-175.

We prove that on a Kähler manifold admitting an extremal metric $\omega$ and for any Kähler potential ${\varphi }_0$ close to $\omega$, the Calabi flow starting at ${\varphi }_0$ exists for all time and the modified Calabi flow starting at ${\varphi }_0$ will always be close to $\omega$. Furthermore, when the initial data is invariant under the maximal compact subgroup of the identity component of the reduced automorphism group, the modified Calabi flow converges to an extremal metric near $\omega$ exponentially fast.

Classification : 53C44, 32Q15, 32Q26

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

[2] E. Calabi, Extremal Kähler metrics, In: “Seminar on Differential Geometry”, Ann. of Math. Stud., Vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, 259–290. | MR | Zbl

[3] E. Calabi, Extremal Kähler metrics. II, In: “Differential Geometry and Complex Analysis”, Springer, Berlin, 1982, 95–114. | MR | Zbl

[4] E. Calabi and X. X. Chen, The space of Kähler metrics. II, J. Differential Geom. 61 (2002), 173–193. | MR | Zbl

[5] X. X. Chen, The space of Kähler metrics, J. Differential Geom. 56 (2000), 189–234. | MR | Zbl

[6] X. X. Chen, Calabi flow in Riemann surfaces revisited: a new point of view, Internat. Math. Res. Notices 2001, 275–297. | MR | Zbl

[7] X. X. Chen, W. Y. Ding and K. Zheng, Pseudo-Calabi flow, Unpublished, 2009.

[8] X. X. Chen and W. Y. He, On the Calabi flow, Amer. J. Math. 130 (2008), 539–570. | MR | Zbl

[9] X. X. Chen and W. Y. He, The Calabi flow on Kähler surface with bounded Sobolev constant, (I), arXiv:0710.5159, 2007. | MR | Zbl

[10] X. X. Chen and W. Y. He, The Calabi flow on toric Fano surface, arXiv:0807.3984, 2008. | MR | Zbl

[11] X. X. Chen, H. Z. Li and B. Wang, Kähler-Ricci flow with small initial energy, Geom. Funct. Anal. 18 (2009), 1525–1563. | MR | Zbl

[12] X. X. Chen and G. Tian, Ricci flow on Kähler manifolds, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), 245–248. | MR | Zbl

[13] X. X. Chen and G. Tian, Ricci flow on Kähler-Einstein surfaces, Invent. Math. 147 (2002), 487–544. | MR | Zbl

[14] X. X. Chen and G. Tian, Uniqueness of extremal Kähler metrics, C. R. Math. Acad. Sci. Paris 340 (2005), 287–290. | MR | Zbl

[15] X. X. Chen and M. J. Zhu, Liouville energy on a topological two sphere, arXiv:0710.4320, 2007. | MR

[16] P. T. Chruściel, Semi-global existence and convergence of solutions of the Robinson-Trautman ($2$-dimensional Calabi) equation, Comm. Math. Phys. 137 (1991), 289–313. | MR | Zbl

[17] S. K. Donaldson, Symmetric spaces, Kähler geometry and Hamiltonian dynamics, In: “Northern California Symplectic Geometry Seminar”, Amer. Math. Soc. Transl. Ser. 2, 196, Amer. Math. Soc., Providence, RI, 1999, 3–33. | MR | Zbl

[18] S. K. Donaldson, Conjectures in Kähler geometry, In: “Strings and Geometry”, Amer. Math. Soc., Providence, RI, 2004, 71–78. | MR | Zbl

[19] W. Y. He, Local solution and extension to the Calabi flow, arXiv:0904.0978, 2009. | MR | Zbl

[20] T. Mabuchi, Some symplectic geometry on compact Kähler manifolds. I, Osaka J. Math. 24 (1987), 227–252. | MR | Zbl

[21] S. Semmes, Complex Monge-Ampère and symplectic manifolds, Amer. J. Math. 114 (1992), 495–550. | MR | Zbl

[22] M. Struwe, Curvature flows on surfaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) (2002), 247–274. | EuDML | Numdam | MR | Zbl

[23] V. Tosatti and B. Weinkove, The Calabi flow with small initial energy, Math. Res. Lett. 14 (2007), 1033–1039. | MR | Zbl

[24] K. Zheng, Stability of the Kähler Ricci flow in the space of Kähler metrics, Pacific J. Math. 251 (2011), 469–497. | MR | Zbl