[Paper Review] Hermitian Curvature Flow

Jeffrey Streets, Gang Tian|ArXiv.org|Apr 25, 2008

Geometry and complex manifolds9 references4 citations

TL;DR

This paper introduces Hermitian Curvature Flow (HCF), a parabolic flow for Hermitian metrics on complex manifolds that evolves the metric to reduce torsion and curvature, preserving the complex structure. It establishes short-time existence, regularity, and stability near Kähler-Einstein metrics with non-positive first Chern class, showing solutions converge to Kähler-Einstein metrics under small initial perturbations.

ABSTRACT

We define a functional for Hermitian metrics using the curvature of the Chern connection. The Euler-Lagrange equation for this functional is an elliptic equation for Hermitian metrics. Solutions to this equation are related to Kähler-Einstein metrics, and are automatically Kähler-Einstein under certain conditions. Given this, a natural parabolic flow equation arises. We prove short time existence and regularity results for this flow, as well as stability for the flow near Kähler-Einstein metrics with negative or zero first Chern class.

Motivation & Objective

• To define a natural parabolic flow for Hermitian metrics on complex manifolds that generalizes Ricci flow in the non-Kähler setting.
• To establish short-time existence and regularity of the flow under curvature and torsion bounds.
• To prove stability of the flow near Kähler-Einstein metrics with non-positive first Chern class.
• To show that the flow preserves the Hermitian condition and reduces torsion, suggesting convergence to Kähler metrics.

Proposed method

• Define a functional involving scalar Chern curvature, torsion norm, and trace of torsion, whose Euler-Lagrange equation motivates the flow.
• Introduce the Hermitian Curvature Flow (HCF) via the evolution equation ∂g/∂t = -S + Q, where S is the Chern-Ricci curvature and Q is a quadratic torsion term.
• Prove short-time existence using a priori estimates and the inverse function theorem in a suitable function space.
• Establish higher-order derivative estimates under curvature bounds, leading to maximal existence time controlled by blow-up of curvature and torsion norms.
• Use Hodge-type operators and complex geometry tools to express the flow in terms of the Kähler form ω(t), showing compatibility with the complex structure.
• Apply integration by parts and curvature identities to derive evolution equations for |w|² and other geometric quantities, enabling energy estimates.

Experimental results

Research questions

• RQ1 Can a parabolic flow be defined on Hermitian manifolds that preserves the complex structure and evolves metrics toward Kähler metrics?
• RQ2 Does the Hermitian Curvature Flow admit short-time existence for arbitrary Hermitian initial metrics?
• RQ3 Is the flow stable near Kähler-Einstein metrics with non-positive first Chern class?
• RQ4 Are static solutions of HCF necessarily Kähler-Einstein?
• RQ5 Can the flow be expressed in terms of the evolving Kähler form ω(t) using Hodge-theoretic operators?

Key findings

• The Hermitian Curvature Flow admits a unique solution for a time interval depending only on the initial curvature and torsion norms, with explicit lower bound c(n)/max{ |Ω|₀, |∇T|₀, |T|²₀ }.
• Higher-order derivative estimates for curvature and torsion hold uniformly in time, with decay rates of order t^{-m/2} for the m-th derivative.
• The solution exists on a maximal time interval [0, τ), and if τ < ∞, then the curvature or torsion norm must blow up.
• The flow is stable near Kähler-Einstein metrics with c₁ ≤ 0: small C∞-perturbations of such metrics lead to global solutions converging to a Kähler-Einstein metric.
• When the initial metric is Kähler, the HCF reduces to Kähler-Ricci flow, preserving the Kähler condition.
• Static solutions of HCF are Kähler-Einstein under certain conditions, suggesting the flow evolves Hermitian metrics toward Kähler-Einstein metrics.