The Calabi functional on a ruled surface

Székelyhidi, Gábor

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 42 (2009) no. 5, pp. 837-856.

On étudie la fonctionnelle de Calabi sur une surface réglée au-dessus d'une courbe de genre deux. Pour les polarizations qui n'admettent pas de métrique extrémale, on décrit le comportement d'une suite minimisante partitionnant la variété. On montre aussi que le flot de Calabi partant d'une métrique avec une symétrie appropriée produit une telle suite minimisante.

We study the Calabi functional on a ruled surface over a genus two curve. For polarizations which do not admit an extremal metric we describe the behavior of a minimizing sequence splitting the manifold into pieces. We also show that the Calabi flow starting from a metric with suitable symmetry gives such a minimizing sequence.

[1] V. Apostolov, D. M. J. Calderbank, P. Gauduchon & C. W. Tønnesen-Friedman, Hamiltonian 2-forms in Kähler geometry. III. Extremal metrics and stability, Invent. Math. 173 (2008), 547-601. | MR | Zbl

[2] T. Aubin, Équations du type Monge-Ampère sur les variétés kählériennes compactes, Bull. Sci. Math. 102 (1978), 63-95. | MR | Zbl

[3] E. Calabi, Extremal Kähler metrics, in Seminar on Differential Geometry, Princeton (S. T. Yau, éd.), 1982. | Zbl

[4] X. Chen, Calabi flow in Riemann surfaces revisited: a new point of view, Int. Math. Res. Not. 2001 (2001), 275-297. | MR | Zbl

[5] X. Chen, Space of Kähler metrics. III. On the lower bound of the Calabi energy and geodesic distance, Invent. Math. 175 (2009), 453-503. | MR | Zbl

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

[7] X. Chen & W. Y. He, The Calabi flow on toric Fano surfaces, preprint arXiv:0807.3984. | MR | Zbl

[8] 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

[9] S. K. Donaldson, Remarks on gauge theory, complex geometry and $4$-manifold topology, in Fields Medallists' lectures, World Sci. Ser. 20th Century Math. 5, World Sci. Publ., River Edge, NJ, 1997, 384-403. | MR

[10] S. K. Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002), 289-349. | MR | Zbl

[11] S. K. Donaldson, Conjectures in Kähler geometry, in Strings and geometry, Clay Math. Proc. 3, Amer. Math. Soc., 2004, 71-78. | MR | Zbl

[12] S. K. Donaldson, Lower bounds on the Calabi functional, J. Differential Geom. 70 (2005), 453-472. | MR | Zbl

[13] A. Fujiki, Moduli space of polarized algebraic manifolds and Kähler metrics, Sugaku Expositions 5 (1992), 173-191. | MR | Zbl

[14] D. Guan, Extremal solitons and exponential $C^{\infty }$ convergence of the modified Calabi flow on certain $ℂ{\mathrm{P}}^1$ bundles, Pacific J. Math. 233 (2007), 91-124. | MR | Zbl

[15] A. D. Hwang, On the Calabi energy of extremal Kähler metrics, Internat. J. Math. 6 (1995), 825-830. | MR | Zbl

[16] A. D. Hwang & M. A. Singer, A momentum construction for circle-invariant Kähler metrics, Trans. Amer. Math. Soc. 354 (2002), 2285-2325. | MR | Zbl

[17] T. Mabuchi, Stability of extremal Kähler manifolds, Osaka J. Math. 41 (2004), 563-582. | MR | Zbl

[18] J. Ross & R. Thomas, An obstruction to the existence of constant scalar curvature Kähler metrics, J. Differential Geom. 72 (2006), 429-466. | MR | Zbl

[19] N. Sesum & G. Tian, Bounding scalar curvature and diameter along the Kähler Ricci flow (after Perelman), J. Inst. Math. Jussieu 7 (2008), 575-587. | MR | Zbl

[20] M. Struwe, Curvature flows on surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. 1 (2002), 247-274. | Numdam | MR | Zbl

[21] G. Székelyhidi, Extremal metrics and $K$-stability, Thèse, Imperial College London, 2006, arXiv:math/0611002. | MR | Zbl

[22] G. Székelyhidi, Extremal metrics and $K$-stability, Bull. Lond. Math. Soc. 39 (2007), 76-84. | MR | Zbl

[23] G. Tian, On Calabi's conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990), 101-172. | Zbl

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

[25] C. W. Tønnesen-Friedman, Extremal Kähler metrics on minimal ruled surfaces, J. reine angew. Math. 502 (1998), 175-197. | Zbl

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

[27] R. Ye, The logarithmic Sobolev inequality along the Ricci flow, preprint arXiv:0707.2424, 2007.

[28] Q. S. Zhang, A uniform Sobolev inequality under Ricci flow, Int. Math. Res. Not. IMRN 17 (2007), Art. ID rnm056. | Zbl