A parabolic flow of balanced metrics

TL;DR

This paper introduces a new flow in balanced geometry, extending the Calabi flow, and proves short-time existence, uniqueness, and properties of solutions, including on the Iwasawa manifold.

Contribution

It generalizes a method for evolution equations to balanced geometry and establishes a natural extension of the Calabi flow with proven short-time existence and stability.

Findings

Flow preserves Bott-Chern cohomology class.

Flow maintains the Kähler condition.

Explicit analysis on the Iwasawa manifold.

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 recently by Bryant and Xu for studying the Laplacian flow in 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 Kaehler condition. Finally we study explicitly the flow on the Iwasawa manifold.