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# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

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1435-5345
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Volume 2018, Issue 739

# Pluriclosed flow on generalized Kähler manifolds with split tangent bundle

Jeffrey Streets
Published Online: 2015-09-17 | DOI: https://doi.org/10.1515/crelle-2015-0055

## Abstract

We show that the pluriclosed flow preserves generalized Kähler structures with the extra condition $\left[{J}_{+},{J}_{-}\right]=0$, 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.

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Revised: 2015-06-02

Published Online: 2015-09-17

Published in Print: 2018-06-01

Funding Source: National Science Foundation

Award identifier / Grant number: DMS-1301864

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

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2018, Issue 739, Pages 241–276, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

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