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The behavior of the second Ricci flow on complex parallelizable manifolds

Lucio Bedulli, Luigi Vezzoni

TL;DR

This work analyzes the Hermitian curvature flow in the form $\partial_t g=-\tilde{R}$ (the second Chern-Ricci flow with $Q=0$) on compact complex parallelizable manifolds. It proves long-time existence under a uniform bound on the torsion-trace $|w|_g$ by deriving $C^1$ and higher-order estimates via a Yau–type $C^3$-style approach, and shows convergence to a Chern-flat metric for diagonal initial data. The paper also establishes dynamical stability of Chern-flat metrics: linearization at such a metric is the Chern Laplacian, whose negative semidefinite spectrum on balanced backgrounds yields exponential decay of the second Chern-Ricci tensor and smooth convergence to a Chern-flat limit. These results extend the understanding of Hermitian curvature flows beyond the Kähler setting and provide attractor behavior toward flat Chern geometry on complex parallelizable manifolds. Collectively, they identify Chern-flat metrics as robust end-states for the flow under perturbations in this geometric context.

Abstract

We study the flow of Hermitian metrics governed by the second Chern-Ricci form on a compact complex manifolds. The flow belongs to the family of Hermitian curvature flows introduced by Streets and Tian and it was considered by Lee in order to study compact Hermitian manifolds with almost negative Chern bisectional curvature. We show a regularity result on compact complex parallelizable manifolds and we prove that Chern-flat metrics are dynamically stable.

The behavior of the second Ricci flow on complex parallelizable manifolds

TL;DR

This work analyzes the Hermitian curvature flow in the form (the second Chern-Ricci flow with ) on compact complex parallelizable manifolds. It proves long-time existence under a uniform bound on the torsion-trace by deriving and higher-order estimates via a Yau–type -style approach, and shows convergence to a Chern-flat metric for diagonal initial data. The paper also establishes dynamical stability of Chern-flat metrics: linearization at such a metric is the Chern Laplacian, whose negative semidefinite spectrum on balanced backgrounds yields exponential decay of the second Chern-Ricci tensor and smooth convergence to a Chern-flat limit. These results extend the understanding of Hermitian curvature flows beyond the Kähler setting and provide attractor behavior toward flat Chern geometry on complex parallelizable manifolds. Collectively, they identify Chern-flat metrics as robust end-states for the flow under perturbations in this geometric context.

Abstract

We study the flow of Hermitian metrics governed by the second Chern-Ricci form on a compact complex manifolds. The flow belongs to the family of Hermitian curvature flows introduced by Streets and Tian and it was considered by Lee in order to study compact Hermitian manifolds with almost negative Chern bisectional curvature. We show a regularity result on compact complex parallelizable manifolds and we prove that Chern-flat metrics are dynamically stable.
Paper Structure (6 sections, 8 theorems, 108 equations)

This paper contains 6 sections, 8 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Uniform Estimates
  4. Proof of Theorem \ref{['main1']}
  5. Stability: proof of Theorem \ref{['stability']}
  6. Remarks

Key Result

Theorem 1.1

Let $g$ be a maximal time solution to the Hermitian curvature flow Q=0 on a compact complex parallelizable manifold. Assume that $|w|_{g}$ is uniformly bounded, then $g$ is defined in $M\times [0,\infty)$.

Theorems & Definitions (21)

  • Theorem 1.1: Long-time existence
  • Theorem 1.2: Stability
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 11 more