A parabolic flow of balanced metrics

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.

19 pages. Revised version. To appear in Crelle's Journal

References in corpus (5)

• On the $\partial \overline{\partial }$-Lemma and Bott-Chern cohomology
• Autour de la cohomologie de Bott-Chern
• Short-time behaviour of a modified Laplacian coflow of G2-structures
• The Chern-Ricci flow on complex surfaces
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Cited by in corpus (8)

• A flow of isometric $G_2$-structures. Short-time existence
• Geometric Partial Differential Equations from Unified String Theories
• On the existence of balanced metrics on six-manifolds of cohomogeneity one
• Estimates for a geometric flow for the Type IIB string
• Parabolic complex Monge-Ampère equations on compact Hermitian manifolds
• Blowing up Chern-Ricci flat balanced metrics
• The continuity equation for Hermitian metrics: Calabi estimates, Chern scalar curvature and Oeljeklaus-Toma manifolds
• Deformation of Hermitian metrics