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An Introduction to the Kähler–Ricci Flow

  • Jian SongEmail author
  • Ben Weinkove
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2086)

Abstract

These notes give an introduction to the Kähler–Ricci flow. We give an exposition of a number of well-known results including: maximal existence time for the flow, convergence on manifolds with negative and zero first Chern class, and behavior of the flow in the case when the canonical bundle is big and nef. We also discuss the collapsing of the Kähler–Ricci flow on the product of a torus and a Riemann surface of genus greater than one. Finally, we discuss the connection between the flow and the minimal model program with scaling, the behavior of the flow on general Kähler surfaces and some other recent results and conjectures.

Keywords

Minimal Model Program (MMP) Parabolic Complex Mori Fiber Space Mabuchi Energy Hermitian Metric 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

Jian Song is supported in by an NSF CAREER grant DMS-08-47524 and a Sloan Research Fellowship. Ben Weinkove is supported by the NSF grants DMS-08-48193 and DMS-11-05373.

References

  1. [Aub78]
    T. Aubin, Equation de type Monge-Ampère sur les variétés kählériennes compactes. Bull. Sci. Math. 102, 63–95 (1978)MathSciNetzbMATHGoogle Scholar
  2. [Aub82]
    T. Aubin, in Nonlinear Analysis on Manifolds. Monge-Ampère Equations. Grundlehren der Mathematischen Wissenschaften, vol. 252 (Springer, New York, 1982)Google Scholar
  3. [Bando84]
    S. Bando, On the classification of three-dimensional compact Kaehler manifolds of nonnegative bisectional curvature. J. Differ. Geom. 19(2), 283–297 (1984)MathSciNetzbMATHGoogle Scholar
  4. [Bando87]
    S. Bando, The K-energy map, almost Einstein Kähler metrics and an inequality of the Miyaoka-Yau type. Tohoku Math. J. 39, 231–235 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [BHPV]
    W.P. Barth, K. Hulek, C. Peters, A. Van de Ven, in Compact Complex Surfaces, 2nd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete (Springer, Berlin, 2004)Google Scholar
  6. [BCHM10]
    C. Birkar, P. Cascini, C. Hacon, J. McKernan, Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23(2), 405–468 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [BW08]
    C. Böhm, B. Wilking, Manifolds with positive curvature operators are space forms. Ann. Math. (2) 167(3), 1079–1097 (2008)Google Scholar
  8. [BEGZ10]
    S. Boucksom, P. Eyssidieux, V. Guedj, A. Zeriahi, Monge-Ampère equations in big cohomology classes. Acta Math. 205, 199–262 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [BS08]
    S. Brendle, R. Schoen, Classification of manifolds with weakly 1 ∕ 4-pinched curvatures. Acta Math. 200(1), 1–13 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [Buch99]
    N. Buchdahl, On compact Kähler surfaces. Ann. Inst. Fourier 49(1), 287–302 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [Cal57]
    E. Calabi, On Kähler manifolds with vanishing canonical class, in Algebraic Geometry and Topology. A Symposium in Honor of S. Lefschetz (Princeton University Press, Princeton, 1957), pp. 78–89Google Scholar
  12. [Cal58]
    E. Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens. Mich. Math. J. 5, 105–126 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [Cal82]
    E. Calabi, Extremal Kähler metrics, in Seminar on Differential Geometry. Annals of Mathematics Studies, vol. 102, (Princeton University Press, Princeton, 1982), pp. 259–290Google Scholar
  14. [Cao85]
    H.D. Cao, Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds. Invent. Math. 81(2), 359–372 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [Cao92]
    H.D. Cao, On Harnack’s inequalities for the Kähler–Ricci flow. Invent. Math. 109(2), 247–263 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [Cao94]
    H.-D. Cao, in Existence of Gradient Kähler–Ricci Solitons. Elliptic and Parabolic Methods in Geometry (Minneapolis, MN, 1994) (A.K. Peters, Wellesley, 1996), pp. 1–16Google Scholar
  17. [Cao]
    H.-D. Cao, The Kähler–Ricci flow on Fano manifolds, in An Introduction to the Kähler–Ricci Flow, ed. by S. Boucksom, P. Eyssidieux, V. Guedj. Lecture Notes in Mathematics (Springer, Heidelberg, 2013)Google Scholar
  18. [CZ09]
    H.-D. Cao, M. Zhu, A note on compact Kähler–Ricci flow with positive bisectional curvature. Math. Res. Lett. 16(6), 935–939 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [CZ06]
    H.-D. Cao, X.-P. Zhu, A complete proof of the Poincaré and geometrization conjectures - application of the Hamilton-Perelman theory of the Ricci flow. Asian J. Math. 10(2), 165–492 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [ChauT06]
    A. Chau, L.-F. Tam, On the complex structure of Kähler manifolds with nonnegative curvature. J. Differ. Geom. 73(3), 491–530 (2006)MathSciNetzbMATHGoogle Scholar
  21. [CST09]
    X. Chen, S. Sun, G. Tian, A note on Kähler–Ricci soliton. Int. Math. Res. Not. IMRN 2009(17), 3328–3336 (2009)MathSciNetzbMATHGoogle Scholar
  22. [CheT06]
    X.X. Chen, G. Tian, Ricci flow on Kähler-Einstein manifolds. Duke Math. J. 131(1), 17–73 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [ChW09]
    X. Chen, B. Wang, The Kähler–Ricci flow on Fano manifolds (I). J. Eur. Math. Soc. (JEMS) 14(6), 2001–2038 (2012)Google Scholar
  24. [ChengYau75]
    S.Y. Cheng, S.T. Yau, Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math. 28(3), 333–354 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  25. [Cher87]
    P. Cherrier, Équations de Monge-Ampère sur les variétés hermitiennes compactes. Bull. Sci. Math. (2) 111(4), 343–385 (1987)Google Scholar
  26. [Chow91]
    B. Chow, The Ricci flow on the 2-sphere. J. Differ. Geom. 33(2), 325–334 (1991)zbMATHGoogle Scholar
  27. [ChowKnopf]
    B. Chow, D. Knopf, in The Ricci Flow: An Introduction. Mathematical Surveys and Monographs, vol. 110 (American Mathematical Society, Providence, 2004), xii + 325 pp.Google Scholar
  28. [CLN06]
    B. Chow, P. Lu, L. Ni, in Hamilton’s Ricci Flow. Graduate Studies in Mathematics, vol. 77 (American Mathematical Society/Science Press, Providence/New York, 2006), xxxvi + 608 pp.Google Scholar
  29. [CKL11]
    A. Corti, A.-S. Kaloghiros, V. Lazić, Introduction to the Minimal Model Program and the existence of flips. Bull. Lond. Math. Soc. 43(3), 415–418 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. [Deb01]
    O. Debarre, in Higher-Dimensional Algebraic Geometry. Universitext (Springer, New York, 2001)Google Scholar
  31. [Dem96]
    J.-P. Demailly, in L 2 Vanishing Theorems for Positive Line Bundles and Adjunction Theory. Transcendental Methods in Algebraic Geometry (Cetraro, 1994). Lecture Notes in Mathematics, vol. 1646 (Springer, Berlin, 1996), pp. 1–97Google Scholar
  32. [DemPaun04]
    J.-P. Demailly, M. Paun, Numerical characterization of the Kähler cone of a compact Kähler manifold. Ann. Math. (2) 159(3), 1247–1274 (2004)Google Scholar
  33. [Det83]
    D.M. DeTurck, Deforming metrics in the direction of their Ricci tensors. J. Differ. Geom. 18 (1), 157–162 (1983)MathSciNetzbMATHGoogle Scholar
  34. [DT92]
    W.-Y. Ding, G. Tian, Kähler-Einstein metrics and the generalized Futaki invariant. Invent. Math. 110(2), 315–335 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  35. [Don02]
    S.K. Donaldson, Scalar curvature and stability of toric varieties. J. Differ. Geom. 62, 289–349 (2002)MathSciNetzbMATHGoogle Scholar
  36. [Eva82]
    L.C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations. Comm. Pure Appl. Math. 35(3), 333–363 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  37. [EGZ11]
    P. Eyssidieux, V. Guedj, A. Zeriahi, Viscosity solutions to degenerate complex Monge-Ampère equations. Comm. Pure Appl. Math. 64, 1059–1094 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  38. [FIK03]
    M. Feldman, T. Ilmanen, D. Knopf, Rotationally symmetric shrinking and expanding gradient Kähler–Ricci solitons. J. Differ. Geom. 65(2), 169–209 (2003)MathSciNetzbMATHGoogle Scholar
  39. [Fo11]
    T.-H.F. Fong, Kähler–Ricci flow on projective bundles over Kähler-Einstein manifolds (2011). Preprint (arXiv:1104.3924 [math.DG])Google Scholar
  40. [GT01]
    D. Gilbarg, N.S. Trudinger, in Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edn. Classics in Mathematics (Springer, Berlin, 2001), xiv + 517 pp.Google Scholar
  41. [Gill11]
    M. Gill, Convergence of the parabolic complex Monge-Ampère equation on compact Hermitian manifolds. Comm. Anal. Geom. 19(2), 277–303 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  42. [GH78]
    P. Griffiths, J. Harris, in Principles of Algebraic Geometry. Pure and Applied Mathematics (Wiley-Interscience, New York, 1978)Google Scholar
  43. [GTZ11]
    M. Gross, V. Tosatti, Y. Zhang, Collapsing of abelian fibred Calabi-Yau manifolds. Duke Math. J. 162(3), 517–551 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  44. [Gu09]
    H.-L. Gu, A new proof of Mok’s generalized Frankel conjecture theorem. Proc. Am. Math. Soc. 137(3), 1063–1068 (2009)CrossRefzbMATHGoogle Scholar
  45. [HM10]
    C.D. Hacon, J. McKernan, Existence of minimal models for varieties of log general type, II. J. Am. Math. Soc. 23(2), 469–490 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  46. [Ham82]
    R.S. Hamilton, Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982)MathSciNetzbMATHGoogle Scholar
  47. [Ham86]
    R.S. Hamilton, Four-manifolds with positive curvature operator. J. Differ. Geom. 24(2), 153–179 (1986)MathSciNetzbMATHGoogle Scholar
  48. [Ham88]
    R. Hamilton, in The Ricci Flow on Surfaces. Mathematics and General Relativity (Santa Cruz, CA, 1986). Contemporary Mathematics, vol. 71 (American Mathematical Society, Providence, 1988), pp. 237–262Google Scholar
  49. [Ham95a]
    R.S. Hamilton, in The Formation of Singularities in the Ricci Flow. Surveys in Differential Geometry, vol. II (Cambridge, MA, 1993) (International Press, Cambridge, 1995), pp. 7–136Google Scholar
  50. [Ha77]
    R. Hartshorne, in Algebraic Geometry. Graduate Texts in Mathematics, vol. 52 (Springer, New York, 1977)Google Scholar
  51. [KMM87]
    Y. Kawamata, K. Matsuda, K. Matsuki, in Introduction to the Minimal Model Problem. Algebraic Geometry (Sendai, 1985). Advanced Studies in Pure Mathematics, vol. 10 (North-Holland, Amsterdam, 1987), pp. 283–360Google Scholar
  52. [KL08]
    B. Kleiner, J. Lott, Notes on Perelman’s papers. Geom. Topol. 12(5), 2587–2855 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  53. [Kob85]
    R. Kobayashi, Einstein-Kähler V-metrics on open Satake V-surfaces with isolated quotient singularities. Math. Ann. 272(3), 385–398 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  54. [KodMor71]
    K. Kodaira, J. Morrow, Complex Manifolds (Holt, Rinehart and Winston, Inc., New York, 1971), vii + 192 pp.Google Scholar
  55. [KolMori98]
    J. Kollár, S. Mori, in Birational Geometry of Algebraic Varieties. Cambridge Tracts in Mathematics, vol. 134 (Cambridge University Press, Cambridge, 1998)Google Scholar
  56. [Kol98]
    S. Kołodziej, The complex Monge-Ampère equation. Acta Math. 180(1), 69–117 (1998); Math. Ann. 342(4), 773–787 (2008); Differ. Geom. 29, 665–683 (1989)Google Scholar
  57. [Kryl82]
    N.V. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations. Izvestia Akad. Nauk. SSSR 46, 487–523 (1982). English translation in Math. USSR Izv. 20(3), 459–492 (1983)Google Scholar
  58. [LaNT09]
    G. La Nave, G. Tian, Soliton-type metrics and Kähler–Ricci flow on symplectic quotients (2009). Preprint (arXiv: 0903.2413 [math.DG])Google Scholar
  59. [Lam99]
    A. Lamari, Le cône Kählérien d’une surface. J. Math. Pure Appl. 78, 249–263 (1999)MathSciNetzbMATHGoogle Scholar
  60. [Laz04]
    R. Lazarsfeld, in Positivity in Algebraic Geometry. I. Classical Setting: Line Bundles and Linear Series. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge. A Series of Modern Surveys in Mathematics, vol. 48 (Springer, Berlin, 2004), xviii + 387 pp.Google Scholar
  61. [Lieb96]
    G.M. Lieberman, Second Order Parabolic Differential Equations (World Scientific, River Edge, 1996)CrossRefzbMATHGoogle Scholar
  62. [Mab86]
    T. Mabuchi, K-energy maps integrating Futaki invariants. Tohoku Math. J. (2) 38(4), 575–593 (1986)Google Scholar
  63. [Mok88]
    N. Mok, The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature. J. Differ. Geom. 27(2), 179–214 (1988)MathSciNetzbMATHGoogle Scholar
  64. [MT07]
    J. Morgan, G. Tian, in Ricci Flow and the Poincaré Conjecture. Clay Mathematics Monographs, vol. 3 (American Mathematical Society/Clay Mathematics Institute, Providence/Cambridge, 2007)Google Scholar
  65. [MT08]
    J. Morgan, G. Tian, Completion of the proof of the geometrization conjecture (2008). Preprint (arXiv: 0809.4040 [math.DG])Google Scholar
  66. [MSz09]
    O. Munteanu, G. Székelyhidi, On convergence of the Kähler–Ricci flow. Commun. Anal. Geom. 19(5), 887–903 (2011)CrossRefzbMATHGoogle Scholar
  67. [Ni04]
    L. Ni, A monotonicity formula on complete Kähler manifolds with nonnegative bisectional curvature. J. Am. Math. Soc. 17(4), 909–946 (2004)CrossRefzbMATHGoogle Scholar
  68. [NiW10]
    L. Ni, B. Wilking, Manifolds with 1/4-pinched flag curvature. Geom. Funct. Anal. 20(2), 571–591 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  69. [Per02]
    G. Perelman, The entropy formula for the Ricci flow and its geometric applications (2002). Preprint (arXiv: math.DG/0211159)Google Scholar
  70. [Per03q]
    G. Perelman, Ricci flow with surgery on three-manifolds (2003). Preprint (arXiv:math.DG/0303109)Google Scholar
  71. [Per03b]
    G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds (2003). Preprint [arXiv:math.DG/0307245]Google Scholar
  72. [PSS07]
    D.H. Phong, N. Sesum, J. Sturm, Multiplier ideal sheaves and the Kähler–Ricci flow. Comm. Anal. Geom. 15(3), 613–632 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  73. [PSSW08b]
    D.H. Phong, J. Song, J. Sturm, B. Weinkove, The Kähler–Ricci flow with positive bisectional curvature. Invent. Math. 173(3), 651–665 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  74. [PSSW09]
    D.H. Phong, J. Song, J. Sturm, B. Weinkove, The Kähler–Ricci flow and the \(\bar{\partial }\) operator on vector fields. J. Differ. Geom. 81(3), 631–647 (2009)MathSciNetzbMATHGoogle Scholar
  75. [PSSW11]
    D.H. Phong, J. Song, J. Sturm, B. Weinkove, On the convergence of the modified Kähler–Ricci flow and solitons. Comment. Math. Helv. 86(1), 91–112 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  76. [PS05]
    D.H. Phong, J. Sturm, On the Kähler–Ricci flow on complex surfaces. Pure Appl. Math. Q. 1(2), Part 1, 405–413 (2005)Google Scholar
  77. [PS06]
    D.H. Phong, J. Sturm, On stability and the convergence of the Kähler–Ricci flow. J. Differ. Geom. 72(1), 149–168 (2006)MathSciNetzbMATHGoogle Scholar
  78. [PS10]
    D.H. Phong, J. Sturm, Lectures on stability and constant scalar curvature, in Handbook of Geometric Analysis, No. 3. Advanced Lectures in Mathematics (ALM), vol. 14 (International Press, Somerville, 2010), pp. 357–436Google Scholar
  79. [Rub09]
    Y. Rubinstein, On the construction of Nadel multiplier ideal sheaves and the limiting behavior of the Ricci flow. Trans. Am. Math. Soc. 361(11), 5839–5850 (2009)CrossRefzbMATHGoogle Scholar
  80. [Se05]
    N. Šešum, Curvature tensor under the Ricci flow. Am. J. Math. 127(6), 1315–1324 (2005)CrossRefzbMATHGoogle Scholar
  81. [SeT08]
    N. Sesum, G. Tian, Bounding scalar curvature and diameter along the Kähler–Ricci flow (after Perelman). J. Inst. Math. Jussieu 7(3), 575–587 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  82. [ShW11]
    M. Sherman, B. Weinkove, Interior derivative estimates for the Kähler–Ricci flow. Pac. J. Math. 257(2), 491–501 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  83. [Shi89]
    W.-X. Shi, Deforming the metric on complete Riemannian manifolds. J. Differ. Geom. 30(1), 223–301 (1989)zbMATHGoogle Scholar
  84. [Shok85]
    V.V. Shokurov, A nonvanishing theorem. Izv. Akad. Nauk SSSR Ser. Mat. 49(3), 635–651 (1985)MathSciNetGoogle Scholar
  85. [Siu87]
    Y.T. Siu, in Lectures on Hermitian-Einstein Metrics for Stable Bundles and Kähler-Einstein Metrics. DMV Seminar, vol. 8 (Birkhäuser, Basel, 1987)Google Scholar
  86. [Siu08]
    Y.-T. Siu, Finite generation of canonical ring by analytic method. Sci. China Ser. A 51(4), 481–502 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  87. [SSW11]
    J. Song, G. Székelyhidi, B. Weinkove, The Kähler–Ricci flow on projective bundles. Int. Math. Res. Not. IMRN 2013(2), 243–257 (2013)Google Scholar
  88. [ST07]
    J. Song, G. Tian, The Kähler–Ricci flow on surfaces of positive Kodaira dimension. Invent. Math. 170(3), 609–653 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  89. [ST12]
    J. Song, G. Tian, Canonical measures and Kähler–Ricci flow. J. Am. Math. Soc. 25, 303–353 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  90. [ST09]
    J. Song, G. Tian, The Kähler–Ricci flow through singularities (2009). Preprint [arXiv:0909.4898]Google Scholar
  91. [ST11]
    J. Song, G. Tian, Bounding scalar curvature for global solutions of the Kähler–Ricci flow (2011). PreprintGoogle Scholar
  92. [SW09]
    J. Song, B. Weinkove, The Kähler–Ricci flow on Hirzebruch surfaces. J. Reine Ange. Math. 659, 141–168 (2011)MathSciNetzbMATHGoogle Scholar
  93. [SW10]
    J. Song, B. Weinkove, Contracting exceptional divisors by the Kähler–Ricci flow. Duke Math. J. 162(2), 367–415 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  94. [SW11]
    J. Song, B. Weinkove, Contracting exceptional divisors by the Kähler–Ricci flow II (2011). Preprint (arXiv:1102.1759 [math.DG])Google Scholar
  95. [SY10]
    J. Song, Y. Yuan, Metric flips with Calabi ansatz. Geom. Funct. Anal. 22(1), 240–265 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  96. [StT10]
    J. Streets, G. Tian, A parabolic flow of pluriclosed metrics. Int. Math. Res. Not. IMRN 2010(16), 3101–3133 (2010)MathSciNetzbMATHGoogle Scholar
  97. [Sz10]
    G. Székelyhidi, The Kähler–Ricci flow and K-stability. Am. J. Math. 132(4), 1077–1090 (2010)CrossRefzbMATHGoogle Scholar
  98. [Tian97]
    G. Tian, Kähler-Einstein metrics with positive scalar curvature. Invent. Math. 130, 239–265 (1997)CrossRefGoogle Scholar
  99. [Tian]
    G. Tian, in Canonical Metrics in Kähler Geometry. Lectures in Mathematics ETH Zürich (Birkhäuser, Basel, 2000)Google Scholar
  100. [Tian02]
    G. Tian, in Geometry and Nonlinear Analysis. Proceedings of the International Congress of Mathematicians, vol. I (Beijing, 2002) (Higher Ed. Press, Beijing, 2002), pp. 475–493Google Scholar
  101. [Tian08]
    G. Tian, New results and problems on Kähler–Ricci flow. Géométrie différentielle, physique mathématique, mathématiques et société, II. Astérisque 322, 71–92 (2008)Google Scholar
  102. [Tzha06]
    G. Tian, Z. Zhang, On the Kähler–Ricci flow on projective manifolds of general type. Chin. Ann. Math. Ser. B 27(2), 179–192 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  103. [TZ07]
    G. Tian, X. Zhu, Convergence of Kähler–Ricci flow. J. Am. Math. Soc. 20(3), 675–699 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  104. [Tos10a]
    V. Tosatti, Kähler–Ricci flow on stable Fano manifolds. J. Reine Angew. Math. 640, 67–84 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  105. [Tos10b]
    V. Tosatti, Adiabatic limits of Ricci-flat Kähler metrics. J. Differ. Geom. 84(2), 427–453 (2010)MathSciNetzbMATHGoogle Scholar
  106. [TWY08]
    V. Tosatti, B. Weinkove, S.-T. Yau, Taming symplectic forms and the Calabi-Yau equation. Proc. Lond. Math. Soc. (3) 97(2), 401–424 (2008)Google Scholar
  107. [Tsu88]
    H. Tsuji, Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type. Math. Ann. 281(1), 123–133 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  108. [Tsu96]
    H. Tsuji, Generalized Bergmann metrics and invariance of plurigenera (1996). Preprint [arXiv:math.CV/9604228]Google Scholar
  109. [Yau78]
    S.T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I. Comm. Pure Appl. Math. 31(3), 339–411 (1978)CrossRefzbMATHGoogle Scholar
  110. [Yau78b]
    S.T. Yau, A general Schwarz lemma for Kähler manifolds. Am. J. Math. 100(1), 197–203 (1978)CrossRefzbMATHGoogle Scholar
  111. [Yau93]
    S.-T. Yau, Open problems in geometry. Proc. Symp. Pure Math. 54, 1–28 (1993)CrossRefGoogle Scholar
  112. [ZhaZha11]
    X. Zhang, X. Zhang, Regularity estimates of solutions to complex Monge-Ampère equations on Hermitian manifolds. J. Funct. Anal. 260(7), 2004–2026 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  113. [Zha06]
    Z. Zhang, On degenerate Monge-Ampère equations over closed Kähler manifolds. Int. Math. Res. Not. Art. ID 63640, 18 pp. (2006)Google Scholar
  114. [Zha09]
    Z. Zhang, Scalar curvature bound for Kähler–Ricci flows over minimal manifolds of general type. Int. Math. Res. Not. IMRN 2009(20), 3901–3912 (2009)zbMATHGoogle Scholar
  115. [Zha10]
    Z. Zhang, Scalar curvature behavior for finite-time singularity of Kähler–Ricci flow. Mich. Math. J. 59(2), 419–433 (2010)CrossRefzbMATHGoogle Scholar
  116. [Zhu07]
    X. Zhu, Kähler–Ricci flow on a toric manifold with positive first Chern class (2007). Preprint [arXiv:math.DG/0703486]Google Scholar

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© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.Department of MathematicsRutgers UniversityPiscatawayUSA
  2. 2.Department of MathematicsUniversity of California San DiegoLa JollaUSA

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