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While the Anomaly flow was originally motivated by string theory, its zero slope case is potentially of considerable interest in non-Kähler geometry, as it is a flow of conformally balanced metrics whose stationary points are precisely Kähler metrics. We establish its convergence on Kähler manifolds for suitable initial data. We also discuss its relation to some current problems in complex geometry.

A flow of conformally balanced metrics with Kähler fixed points