A scalar Calabi-type flow in Hermitian geometry: short-time existence and stability

Abstract

We introduce a new geometric flow of Hermitian metrics which evolves an initial metric along the second derivative of the Chern scalar curvature. The flow depends on the choice of a background metric, it always reduces to a scalar equation and preserves some special classes of Hermitian structures, such as balanced and Gauduchon metrics. We show that the flow has always a unique short-time solution and we provide a stability result when the background metric is Kähler with constant scalar curvature (cscK). The main theorem is obtained by proving a general result about stability of parabolic flows on Riemannian manifolds which is interesting in its own right and in particular implies the stability of the classical Calabi flow near cscK metrics.

Stability of geometric flows of closed forms

“…Here we recall the theorem by Chen and He about the stability of the Calabi-flow. The Calabi-flow was generalized to the context of balanced geometry by the authors in [2] (see also [3] for a generalizations in a different direction). A Hermitian metric on a complex manifold is called balanced if its fundamental form is co-closed (instead of closed as in the Kähler case).…”

“…In this last section we prove theorem 2.1. The scheme of the proof resembles the one of the main theorem of [18] and of [3].…”

The continuity equation for Hermitian metrics: Calabi estimates, Chern scalar curvature, and Oeljeklaus–Toma manifolds

“…If the initial data are Hermitian but not necessarily Kähler and $$\operatorname{Ric}\omega$$ is interpreted as the Ricci curvature of the Chern connection, then Equation (1.1) defines the Chern–Ricci flow introduced by Gill [21]. There is a plethora of the literature on the study of non‐Kähler flows [1, 4, 7, 8, 10, 14, 15, 17–19, 27, 31, 41, 44, 45, 53, 56–58, 60, 61].…”