Surveys in Differential Geometry

Volume 22 (2017)

New curvature flows in complex geometry

Pages: 331 – 364



Duong H. Phong (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)

Sebastien Picard (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)

Xiangwen Zhang (Department of Mathematics, University of California at Irvine)


The Anomaly flow is a flow of $(2, 2)$-forms on a $3$-fold which was originally motivated by string theory and the need to preserve the conformally balanced property of a Hermitian metric in the absence of a $\partial \overline{\partial}$-Lemma. It has revealed itself since to be a remarkable higher order extension of the Ricci flow. It has also led to several other curvature flows which may be interesting from the point of view of both non-Kähler geometry and the theory of non-linear partial differential equations. This is a survey of what is known at the present time about these new curvature flows.

Work supported in part by the National Science Foundation Grants DMS-12-66033 and DMS-1605968, and the Simons Collaboration Grant-523313.

Published 13 September 2018