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Pluriclosed flow on generalized Kähler manifolds with split tangent bundle

  • Jeffrey Streets EMAIL logo

Abstract

We show that the pluriclosed flow preserves generalized Kähler structures with the extra condition [J+,J-]=0, a condition referred to as “split tangent bundle.” Moreover, we show that in this case the flow reduces to a nonconvex fully nonlinear parabolic flow of a scalar potential function. We prove a number of a priori estimates for this equation, including a general estimate in dimension n=2 of Evans–Krylov type requiring a new argument due to the nonconvexity of the equation. The main result is a long-time existence theorem for the flow in dimension n=2, covering most cases. We also show that the pluriclosed flow represents the parabolic analogue to an elliptic problem which is a very natural generalization of the Calabi conjecture to the setting of generalized Kähler geometry with split tangent bundle.

Award Identifier / Grant number: DMS-1301864

Funding statement: The author was partly supported by the National Science Foundation DMS-1301864 and an Alfred P. Sloan Fellowship.

Acknowledgements

The author would like to thank Vestislav Apostolov, Marco Gualtieri and Jess Boling for helpful conversations on the results of this paper. Also, in 2012 after [Nuc. Phys. B 858 (2012), 366–376] was written, the author had a discussion with G. Tian, M. Roček and C. Hull during which it was noted that the results of [Nuclear Physics B 248 (1984), 157–186], [J. High Energy Phys. 2010 (2010), paper no. 060], [Comm. Math. Phys. 269 (2007), 833–849] suggested that the generalized Kähler–Ricci flow of [Nuc. Phys. B 858 (2012), 366–376] should reduce to a potential function. Based on this the author and Tian made preliminary investigations in this direction. The author would like to thank all three for these inspirational discussions. The author also thanks an anonymous referee for helpful comments.

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Received: 2014-10-7
Revised: 2015-6-2
Published Online: 2015-9-17
Published in Print: 2018-6-1

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