×

Kähler-Ricci flow on blowups along submanifolds. (English) Zbl 1428.53102

Summary: In this short note, we study the behavior of Kähler-Ricci flow on Kähler manifolds which contract divisors to smooth submanifolds. We show that the Kähler potentials are Hölder continuous and the flow converges sequentially in Gromov-Hausdorff topology to a compact metric space which is homeomorphic to the base manifold.

MSC:

53E20 Ricci flows
53E30 Flows related to complex manifolds (e.g., Kähler-Ricci flows, Chern-Ricci flows)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Cao, H.-D.: Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds. Invent. Math. 81(2), 359-372 (1985) · Zbl 0574.53042 · doi:10.1007/BF01389058
[2] Datar, V., Guo, B., Song, J., Wang, X.: Connecting toric manifolds by conical Kähler-Einstein metrics. Adv. Math. 323, 38-83 (2018) · Zbl 1384.32020 · doi:10.1016/j.aim.2017.10.035
[3] Fu, X., Guo, B., Song, J.: Geometric estimates for complex Monge-Ampère equations. Journal für die reine und angewandte Mathematik (Crelles Journal). arXiv:1706.01527(accepted) · Zbl 1447.35079
[4] Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics, vol. 152. Birkhauser, Boston (1999). xx+585 pp · Zbl 0953.53002
[5] Gross, M., Tosatti, V., Zhang, Y.: Collapsing of abelian fibered Calabi-Yau manifolds. Duke Math. J. 162(3), 517-551 (2013) · Zbl 1276.32020 · doi:10.1215/00127094-2019703
[6] Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255-306 (1982) · Zbl 0504.53034 · doi:10.4310/jdg/1214436922
[7] Han, Q., Lin, F.: Elliptic Partial Differential Equations, 2nd edn. Courant Lecture Notes in Mathematics, vol. 1. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence (2011). x+147 pp · Zbl 1210.35031
[8] Li, Y.: On collapsing Calabi-Yau fibrations. arXiv:1706.10250(preprint) · Zbl 1470.32072
[9] Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159(preprint) · Zbl 1130.53001
[10] Phong, D. H.; Sturm, Jacob, Chapter Sixteen. On the Singularities of the Pluricomplex Green’s Function, 419-435 (2014), Princeton · Zbl 1329.32020 · doi:10.1515/9781400848935-017
[11] Phong, D.H., Sesum, S., Sturm, J.: Multiplier ideal sheaves and the Kähler-Ricci flow. Commun. Anal. Geom. 15(3), 613-632 (2007) · Zbl 1143.53064 · doi:10.4310/CAG.2007.v15.n3.a7
[12] Phong, D.H., Song, J., Sturm, J., Weinkove, B.: The Kähler-Ricci flow and the \[\bar{\partial } \]∂¯-operator on vector fields. J. Differ. Geom. 81(3), 631-647 (2009) · Zbl 1162.32014 · doi:10.4310/jdg/1236604346
[13] Phong, D.H., Picard, S., Zhang, X.W.: New curvature flows in complex geometry. Surv. Differ. Geom. 22 (2017). arXiv:1806.11235(to appear) · Zbl 1408.32027 · doi:10.4310/SDG.2017.v22.n1.a13
[14] Phong, D.H., Picard, S., Zhang, X.W.: Geometric flows and Strominger systems. Math. Z. 288(1-2), 101-113 (2018) · Zbl 1407.32011 · doi:10.1007/s00209-017-1879-y
[15] Rong, X., Zhang, Y.: Continuity of extremal transitions and flops for Calabi-Yau manifolds. Appendix B by Mark Gross. J. Differ. Geom. 89(2), 233-269 (2011) · Zbl 1264.32021 · doi:10.4310/jdg/1324477411
[16] Song, J.: Ricci flow and birational surgery. arXiv:1304.2607(preprint)
[17] Song, J.: Finite time extinction of the Kähler-Ricci flow. Math. Res. Lett. 21(6), 1435-1449 (2014) · Zbl 1319.53076 · doi:10.4310/MRL.2014.v21.n6.a12
[18] Song, J., Tian, G.: The Kähler-Ricci flow on surfaces of positive Kodaira dimension. Invent. Math. 170(3), 609-653 (2007) · Zbl 1134.53040 · doi:10.1007/s00222-007-0076-8
[19] Song, J., Tian, G.: The Kähler-Ricci flow through singularities. Invent. Math. 207(2), 519-595 (2017) · Zbl 1440.53116 · doi:10.1007/s00222-016-0674-4
[20] Song, J., Weinkove, B.: Contracting exceptional divisors by the Kähler-Ricci flow. Duke Math. J. 162(2), 367-415 (2013) · Zbl 1266.53063 · doi:10.1215/00127094-1962881
[21] Song, J., Weinkove, B.: Contracting exceptional divisors by the Kähler-Ricci flow II. Proc. Lond. Math. Soc. (3) 108(6), 1529-1561 (2014) · Zbl 1301.53066 · doi:10.1112/plms/pdt059
[22] Song, J., Yuan, Y.: Metric flips with Calabi ansatz. Geom. Funct. Anal. 22(1), 240-265 (2012) · Zbl 1248.53057 · doi:10.1007/s00039-012-0151-1
[23] Song, J., Székelyhidi, G., Weinkove, B.: The Kähler-Ricci flow on projective bundles. Int. Math. Res. Not. IMRN 2, 243-257 (2013) · Zbl 1315.53077 · doi:10.1093/imrn/rnr265
[24] Streets, J., Tian, G.: A parabolic flow of pluriclosed metrics. Int. Math. Res. Not. IMRN 16, 3101-3133 (2010) · Zbl 1198.53077
[25] Streets, J., Tian, G.: Hermitian curvature flow. J. Eur. Math. Soc. (JEMS) 13(3), 601-634 (2011) · Zbl 1214.53055 · doi:10.4171/JEMS/262
[26] Tian, G., Zhang, Z.: On the Kähler-Ricci flow on projective manifolds of general type. Chin. Ann. Math. Ser. B 27, 179-192 (2006) · Zbl 1102.53047 · doi:10.1007/s11401-005-0533-x
[27] Tosatti, V., Weinkove, B.: On the evolution of a Hermitian metric by its Chern-Ricci form. J. Differ.Geom. 99(1), 125-163 (2015) · Zbl 1317.53092 · doi:10.4310/jdg/1418345539
[28] Tosatti, V., Zhang, Y.: Finite time collapsing of the Kähler-Ricci flow on threefolds. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18(1), 105-118 (2018) · Zbl 1391.53080
[29] Tosatti, V., Weinkove, B., Yang, X.: The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits. Am. J. Math. 140(3), 653-698 (2018) · Zbl 1401.53055 · doi:10.1353/ajm.2018.0016
[30] Yau, S.-T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I. Commun. Pure Appl. Math. 31(3), 339-411 (1978) · Zbl 0369.53059 · doi:10.1002/cpa.3160310304
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.