Energy functionals and canonical Kähler metrics

Duke Mathematical Journal

Volumn 137, Issue 1, 2007, Pages 159-184

  Song, Jian ^a   Weinkove, Ben ^b  

^a JOHNS HOPKINS UNIVERSITY   (United States)

^b HARVARD UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33947643532     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/S0012-7094-07-13715-3     Document Type: Article

References (36)

1. T. AUBIN, Équations du type Mange-Ampère sur les variétés kähleriennes compactes, Bull. Sci. Math. (2) 102 (1978), 63-95. MR 0494932
2. S. BANDO, The K-energy map, almost Einstein Kähler metrics and an inequality of the Miyaoka-Yau type, Tohoku Math. J. 39 (1987), 231-235. MR 0887939
3. S. BANDO and T. MABUCHI, "Uniqueness of Einstein Kähler metrics modulo connected group actions" in Algebraic Geometry (Sendai, Japan, 1985), Adv. Stud. Pure Math. 10, North-Holland, Amsterdam, 1987, 11-40. MR 0946233
4. X. X. CHEN, On the lower bound of the Mabuchi energy and its application, Internat. Math. Res. Notices 2000, no. 12, 607-623. MR 1772078
5. 1, I: A stability theorem on the Kähler Ricci flow, J. Geom. Anal. 16 (2006), 23-38. MR 2211330
6. _, private communication, 2005.
7. X. X. CHEN and O. TIAN, Ricci flow on Kähler-Einstein surfaces, Invent. Math. 147 (2002), 487-544. MR 1893004
8. _, Ricci flow on Kähler-Einstein manifolds, Duke Math. J. 131 (2006), 17-73. MR 2219236
9. _, Geometry of Kähler metrics and foliations by holomorphic discs, preprint, arXiv:math.DG/0507148
10. P. DELIGNE, "Le determinant de la cohomologie" in Current Trends in Arithmetical Algebraic Geometry (Arcata, Calif., 1985), Contemp. Math. 67, Amer. Math. Soc., Providence, 1987, 93-177. MR 0902592
11. W. Y. DING, Remarks on the existence problem of positive Kähler-Einstein metrics, Math. Ann. 282 (1988), 463-471. MR 0967024
12. W. Y. DING and G. TIAN, Kähler-Einstein metrics and the generalized Futaki invariant, Invent. Math. 110 (1992), 315-335. MR 1185586
13. S. K. DONALDSON, Scalar curvature and projective embeddings, I, J. Differential Geom. 59 (2001), 479-522. MR 1916953
14. _, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002), 289-349. MR 1988506
15. _, Scalar curvature and projective embeddings, II, Q. J. Math. 56 (2005), 345-356. MR 2161248
16. A. FUTAKI, An obstruction to the existence of Einstein Kähler metrics, Invent. Math. 73 (1983), 437-443. MR 0718940
17. A. D. HWANG and G. MASCHLER, Central Kähler metrics with non-constant central curvature, Trans. Amer. Math. Soc. 355 (2003), 2183-2203. MR 1973987
18. T. MABUCHI, K-energy maps integrating Futaki invariants, Tohoku Math. J. (2) 38 (1986), 575-593. MR 0867064
19. G. MASCHLER, Central Kähler metrics, Trans. Amer. Math. Soc. 355 (2003), 2161-2182. MR 1973986
20. S. T. PAUL and G. TIAN, Analysis of geometric stability. Int. Math. Res. Nut. 2004, no. 48, 2555-2591. MR 2078110
21. D. H. PHONG and J. STURM, Stability, energy functionals, and Kähler-Einstein metrics, Comm. Anal. Geom. 11 (2003), 565-597. MR 2015757
22. _, Scalar curvature, moment maps and the Deligne pairing, Amer. J. Math. 126 (2004), 693 - 712. MR 2058389
23. _, The Futaki invariant and the Mabuchi energy of a complete intersection, Comm. Anal. Geom. 12 (2004), 321-343. MR 2074881
24. _, On stability and the convergence of the Kähler-Ricci flow, J. Differential Geom. 72 (2006), 149-168. MR 2215459
25. J. ROSS and R. P. THOMAS, A study of the Hilbert-Mumford criterion for the stability of projective varieties, preprint, arXiv:math.AG/0412519
26. Y. T. SIU, Lectures on Hermitian-Einstein Metrics for Stable Bundles and Kähler-Einstein Metrics, DMV Sem. 8, Birkhäuser, Basel, 1987, MR 0904673
27. J. SONG and B. WEINKOVE, On the convergence and singularities of the J-flow with applications to the Mabuchi energy, preprint, arXiv:math.DG/0410418
28. J. STURM, personal communication, 2002.
29. G. TIAN, The K-energy on hypersurfaces and stability, Comm. Anal. Geom. 2 (1994), 239-265. MR 1312688
30. _, Kähler-Einstein metrics with positive scalar cumature, Invent. Math. 130 (1997), 1-37. MR 1471884
31. _, Canonical Metrics in Kähler Geometry, Lectures Math. ETH Zürich, Birkhäuser, Basel, 2000. MR 1787650
32. G. TIAN and X. ZHU, A nonlinear inequality of Moser-Trudinger type, Calc. Var. Partial Differential Equations 10 (2000), 349-354. MR 1767718
33. B. WEINKOVE, On the J-flow in higher dimensions and the lower boundedness of the Mabuchi energy, J. Differential Geom. 73 (2006), 351-358. MR 2226957
34. S.-T. YAU, On the Ricci curvature of a compact Kähler manifold and the complex Mange-Ampère equation, I, Comm. Pure Appl. Math. 31 (1978), 339-411. MR 0480350
35. _, "Open problems in geometry" in Differential Geometry: Partial Differential Equations on Manifolds (Los Angeles, 1990), Proc. Sympos. Pure Math. 54, Part I, Amer. Math. Soc., Providence, 1993, 1-28. MR 1216573
36. S. ZHANG, Heights and reductions of semi-stable varieties, Compositio Math. 104 (1996), 77-105. MR 1420712