Longtime existence of the Kähler-Ricci flow on $\mathbb {C}^n$
HTML articles powered by AMS MathViewer
- by Albert Chau, Ka-Fai Li and Luen-Fai Tam PDF
- Trans. Amer. Math. Soc. 369 (2017), 5747-5768 Request permission
Abstract:
We produce longtime solutions to the Kähler-Ricci flow for complete Kähler metrics on $\mathbb {C}^n$ without assuming the initial metric has bounded curvature, thus extending results in an earlier work of the authors. We prove the existence of a longtime bounded curvature solution emerging from any complete $U(n)$-invariant Kähler metric with non-negative holomorphic bisectional curvature, and that the solution converges as $t\to \infty$ to the standard Euclidean metric after rescaling. We also prove longtime existence results for more general Kähler metrics on $\mathbb {C}^n$ which are not necessarily $U(n)$-invariant.References
- Huai Dong Cao, Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds, Invent. Math. 81 (1985), no. 2, 359–372. MR 799272, DOI 10.1007/BF01389058
- Huai Dong Cao, On Harnack’s inequalities for the Kähler-Ricci flow, Invent. Math. 109 (1992), no. 2, 247–263. MR 1172691, DOI 10.1007/BF01232027
- Albert Chau, Ka-Fai Li, and Luen-Fai Tam, Deforming complete Hermitian metrics with unbounded curvature, Asian J. Math. 20 (2016), no. 2, 267–292. MR 3480020, DOI 10.4310/AJM.2016.v20.n2.a3
- Albert Chau and Luen-Fai Tam, A survey on the Kähler-Ricci flow and Yau’s uniformization conjecture, Surveys in differential geometry. Vol. XII. Geometric flows, Surv. Differ. Geom., vol. 12, Int. Press, Somerville, MA, 2008, pp. 21–46. MR 2488949, DOI 10.4310/SDG.2007.v12.n1.a2
- Albert Chau and Luen-Fai Tam, On a modified parabolic complex Monge-Ampère equation with applications, Math. Z. 269 (2011), no. 3-4, 777–800. MR 2860264, DOI 10.1007/s00209-010-0760-z
- Esther Cabezas-Rivas and Burkhard Wilking, How to produce a Ricci flow via Cheeger-Gromoll exhaustion, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 12, 3153–3194. MR 3429162, DOI 10.4171/JEMS/582
- Bing-Long Chen and Xi-Ping Zhu, Uniqueness of the Ricci flow on complete noncompact manifolds, J. Differential Geom. 74 (2006), no. 1, 119–154. MR 2260930
- Bing-Long Chen and Xi-Ping Zhu, Volume growth and curvature decay of positively curved Kähler manifolds, Q. J. Pure Appl. Math. 1 (2005), no. 1, 68–108. MR 2154333
- Xu-Qian Fan, A uniqueness result of Kähler Ricci flow with an application, Proc. Amer. Math. Soc. 135 (2007), no. 1, 289–298. MR 2280196, DOI 10.1090/S0002-9939-06-08510-8
- Gregor Giesen and Peter M. Topping, Existence of Ricci flows of incomplete surfaces, Comm. Partial Differential Equations 36 (2011), no. 10, 1860–1880. MR 2832165, DOI 10.1080/03605302.2011.558555
- S. Huang and L.-F. Tam, Kähler-Ricci flow with unbounded curvature, preprint, arXiv:1506.00322.
- Herbert Koch and Tobias Lamm, Geometric flows with rough initial data, Asian J. Math. 16 (2012), no. 2, 209–235. MR 2916362, DOI 10.4310/AJM.2012.v16.n2.a3
- Lei Ni, Ancient solutions to Kähler-Ricci flow, Math. Res. Lett. 12 (2005), no. 5-6, 633–653. MR 2189227, DOI 10.4310/MRL.2005.v12.n5.a3
- Lei Ni and Luen-Fai Tam, Kähler-Ricci flow and the Poincaré-Lelong equation, Comm. Anal. Geom. 12 (2004), no. 1-2, 111–141. MR 2074873
- Lei Ni and Luen-Fai Tam, Poincaré-Lelong equation via the Hodge-Laplace heat equation, Compos. Math. 149 (2013), no. 11, 1856–1870. MR 3133296, DOI 10.1112/S0010437X12000322
- Wan-Xiong Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 (1989), no. 1, 223–301. MR 1001277
- Wan-Xiong Shi, Ricci flow and the uniformization on complete noncompact Kähler manifolds, J. Differential Geom. 45 (1997), no. 1, 94–220. MR 1443333
- Miles Simon, Deformation of $C^0$ Riemannian metrics in the direction of their Ricci curvature, Comm. Anal. Geom. 10 (2002), no. 5, 1033–1074. MR 1957662, DOI 10.4310/CAG.2002.v10.n5.a7
- Oliver C. Schnürer, Felix Schulze, and Miles Simon, Stability of Euclidean space under Ricci flow, Comm. Anal. Geom. 16 (2008), no. 1, 127–158. MR 2411470
- Oliver C. Schnürer, Felix Schulze, and Miles Simon, Stability of hyperbolic space under Ricci flow, Comm. Anal. Geom. 19 (2011), no. 5, 1023–1047. MR 2886716, DOI 10.4310/CAG.2011.v19.n5.a8
- Luen-Fai Tam, Exhaustion functions on complete manifolds, Recent advances in geometric analysis, Adv. Lect. Math. (ALM), vol. 11, Int. Press, Somerville, MA, 2010, pp. 211–215. MR 2648946
- P. M. Topping, Ricci flows with unbounded curvature, Proceedings of the International Congress of Mathematicians, Seoul 2014, arXiv:1408.6866.
- Hung-Hsi Wu and Fangyang Zheng, Examples of positively curved complete Kähler manifolds, Geometry and analysis. No. 1, Adv. Lect. Math. (ALM), vol. 17, Int. Press, Somerville, MA, 2011, pp. 517–542. MR 2882437
- Bo Yang, On a problem of Yau regarding a higher dimensional generalization of the Cohn-Vossen inequality, Math. Ann. 355 (2013), no. 2, 765–781. MR 3010146, DOI 10.1007/s00208-012-0803-3
- Bo Yang and Fangyang Zheng, $U(n)$-invariant Kähler-Ricci flow with non-negative curvature, Comm. Anal. Geom. 21 (2013), no. 2, 251–294. MR 3043747, DOI 10.4310/CAG.2013.v21.n2.a1
Additional Information
- Albert Chau
- Affiliation: Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, British Columbia V6T 1Z2, Canada
- MR Author ID: 749289
- Email: chau@math.ubc.ca
- Ka-Fai Li
- Affiliation: Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, British Columbia V6T 1Z2, Canada
- Email: kfli@math.ubc.ca
- Luen-Fai Tam
- Affiliation: The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, People’s Republic of China
- MR Author ID: 170445
- Email: lftam@math.cuhk.edu.hk
- Received by editor(s): September 23, 2014
- Received by editor(s) in revised form: August 5, 2015, and January 14, 2016
- Published electronically: April 24, 2017
- Additional Notes: The research of the first author was partially supported by NSERC grant no. #327637-06
The research of the third author was partially supported by Hong Kong RGC General Research Fund #CUHK 14305114 - © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 5747-5768
- MSC (2010): Primary 53C55, 58J35
- DOI: https://doi.org/10.1090/tran/6902
- MathSciNet review: 3646777