Abstract
We consider the local solution of the Calabi flow for rough initial data. In particular, we prove that for any smooth metric, there is a C α neighborhood such that the Calabi flow has a short time solution for any C α metric in the neighborhood. We also prove that on a compact Kähler surface, if the evolving metrics of the Calabi flow are all L ∞ equivalent, then the Calabi flow exists for all time and converges to an extremal metric subsequently.
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Communicated by Jiaping Wang.
This work was done when the author was a PIMS postdoctoral at University of British Columbia.
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He, W. Local Solution and Extension to the Calabi Flow. J Geom Anal 23, 270–282 (2013). https://doi.org/10.1007/s12220-011-9247-3
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DOI: https://doi.org/10.1007/s12220-011-9247-3