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Transactions of the American Mathematical Society

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Longtime existence of the Kähler-Ricci flow on $ \mathbb{C}^n$


Authors: Albert Chau, Ka-Fai Li and Luen-Fai Tam
Journal: Trans. Amer. Math. Soc. 369 (2017), 5747-5768
MSC (2010): Primary 53C55, 58J35
DOI: https://doi.org/10.1090/tran/6902
Published electronically: April 24, 2017
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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.


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Additional Information

Albert Chau
Affiliation: Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, British Columbia V6T 1Z2, Canada
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
Email: lftam@math.cuhk.edu.hk

DOI: https://doi.org/10.1090/tran/6902
Keywords: K\"ahler-Ricci flow, $U(n)$-invariant K\"ahler metrics
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
Article copyright: © Copyright 2017 American Mathematical Society

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