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# Surveys in Differential Geometry

## Volume 22 (2017)

### New curvature flows in complex geometry

Pages: 331 – 364

DOI: http://dx.doi.org/10.4310/SDG.2017.v22.n1.a13

#### Authors

#### Abstract

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