A parabolic flow of balanced metrics

Abstract

Abstract We prove a general criterion to establish existence and uniqueness of a short-time solution to an evolution equation involving “closed” sections of a vector bundle, generalizing a method used by Bryant and Xu [8] for studying the Laplacian flow in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>G</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> $G_{2}$ -geometry. We apply this theorem in balanced geometry introducing a natural extension of the Calabi flow to the balanced case. We show that this flow has always a unique short-time solution belonging to the same Bott–Chern cohomology class of the initial balanced structure and that it preserves the Kähler condition. Finally, we study explicitly the flow on the Iwasawa manifold.