Abstract
In this note, we study a Kähler–Ricci flow modified from the classic version. In the non-degenerate case, strong convergence at infinite time is achieved. The main focus should be on degenerate case, where some partial results are presented.
Similar content being viewed by others
References
Cao H.: Deformation of Kaehler metrics to Kaehler-Einstein metrics on compact Kaehler manifolds. Invent. Math. 81(2), 359–372 (1985)
Eyssidieux, P., Guedj, V., Zeriahi, A.: Singular Kähler-Einstein metrics. ArXiv:math/0603431 (math.AG) (math.DG)
David, G., Neil, T.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Grundlehren der Mathematischen Wissenschaften, p. 224
Griffiths, P., Harris, J.: Principles of algebraic geometry. Pure and Applied Mathematics, p. xii+813. Wiley-Interscience [Wiley], New York (1978)
Hamilton R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982)
Kawamata Y.: The cone of curves of algebraic varieties. Ann. Math. (2) 119(3), 603–633 (1984)
Kawamata Y.: A generalization of Kodaira-Ramanujam’s vanishing theorem. Math. Ann. 261(1), 43–46 (1982)
Kolodziej, S.: The complex Monge-Ampère equation and pluripotential theory. Mem. Am. Math. Soc. 178(840), pp. x+64 (2005)
Kolodziej S.: Hölder continuity of solutions to the complex Monge-Ampère equation with the right-hand side in L p: the case of compact Kähler manifolds. Math. Ann. 342(2), 379–386 (2008)
Song J., Tian G.: The Kähler–Ricci flow on surfaces of positive Kodaira dimension. Invent. Math. 170(3), 609–653 (2007)
Tian, G., Zhang, Z.: On the Kähler–Ricci flow on projective manifolds of general type. Chin. Ann. Math. Ser. B 27(2), 179–192
Tosatti, V.: Limits of Calabi-Yau metrics when the Kahler class degenerates. ArXiv:0710.4579 (math.DG) (math.AG)
Tsuji H.: Existence and degeneration of Kaehler-Einstein metrics on minimal algebraic varieties of general type. Math. Ann. 281(1), 123–133 (1988)
Tsuji, H.: Degenerate Monge-Ampère equation in algebraic geometry. Miniconference on analysis and applications (Brisbane, 1993). In: Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 33, pp. 209–224. Austral. Nat. Univ., Canberra (1994)
Yau S.T.: On the Ricci curvature of a compact Kaehler manifold and the complex Monge-Ampère equation I. Comm. Pure Appl. Math. 31(3), 339–411 (1978)
Zhang, Z.: On degenerate Monge-Ampère equations over closed Kähler Manifolds. Int. Math. Res. Not. 2006, Art. ID 63640, p. 18
Zhang, Z.: Degenerate Monge-Ampère equations over projective manifolds. Ph.D. Thesis. MIT (2006)
Zhang, Z.: Scalar curvature bound for Kähler–Ricci flows over minimal manifolds of general type. ArXiv:0801.3248 (math.DG) (Submitted)
Zhang, Z.: Scalar curvature behavior for finite time singularity of Kähler–Ricci flow. ArXiv: 0901.1474 (math.DG). Submitted
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, Z. A modified Kähler–Ricci flow. Math. Ann. 345, 559–579 (2009). https://doi.org/10.1007/s00208-009-0365-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-009-0365-1