Pluriclosed flow on generalized Kähler manifolds with split tangent bundle

Pluriclosed flow on generalized Kähler manifolds with split tangent bundle AbstractWe show that the pluriclosed flow preserves generalizedKähler structures with the extra condition [J+,J-]=0[J_{+},J_{-}]=0, a conditionreferred to as “split tangent bundle.” Moreover, we showthat in this case the flow reduces to a nonconvex fully nonlinearparabolic flow of a scalar potential function. We prove a number of a prioriestimates for this equation, including a general estimate in dimension n=2n=2ofEvans–Krylov type requiring a new argument due to the nonconvexity of theequation. The main result is a long-time existence theorem for the flowin dimension n=2n=2, covering most cases. We also show that the pluriclosed flowrepresents the parabolic analogue to an elliptic problem which is a very naturalgeneralization of the Calabi conjecture to the setting of generalized Kählergeometry with split tangent bundle. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal für die reine und angewandte Mathematik de Gruyter

Pluriclosed flow on generalized Kähler manifolds with split tangent bundle

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Publisher
De Gruyter
Copyright
© 2018 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1435-5345
eISSN
1435-5345
D.O.I.
10.1515/crelle-2015-0055
Publisher site
See Article on Publisher Site

Abstract

AbstractWe show that the pluriclosed flow preserves generalizedKähler structures with the extra condition [J+,J-]=0[J_{+},J_{-}]=0, a conditionreferred to as “split tangent bundle.” Moreover, we showthat in this case the flow reduces to a nonconvex fully nonlinearparabolic flow of a scalar potential function. We prove a number of a prioriestimates for this equation, including a general estimate in dimension n=2n=2ofEvans–Krylov type requiring a new argument due to the nonconvexity of theequation. The main result is a long-time existence theorem for the flowin dimension n=2n=2, covering most cases. We also show that the pluriclosed flowrepresents the parabolic analogue to an elliptic problem which is a very naturalgeneralization of the Calabi conjecture to the setting of generalized Kählergeometry with split tangent bundle.

Journal

Journal für die reine und angewandte Mathematikde Gruyter

Published: Jun 1, 2018

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