[Paper Review] The Kähler-Ricci flow on manifolds with negative holomorphic curvature

Freid Tong|arXiv (Cornell University)|May 9, 2018

Geometry and complex manifolds12 references3 citations

TL;DR

This paper establishes the long-time existence and smooth convergence of the normalized Kähler-Ricci flow on complete noncompact Kähler manifolds with negative holomorphic sectional curvature. Using elementary maximum principle techniques, it proves convergence to a complete Kähler-Einstein metric of negative scalar curvature, recovering a result of Wu and Yau without relying on advanced analytic tools from the Kähler cone or quasi-bounded coordinate charts.

ABSTRACT

We study the behaviour of the normalized Kähler-Ricci flow on complete Kähler manifolds of negative holomorphic sectional curvature. We show that the flow exists for all time and converges to a Kähler-Einstein metric of negative scalar curvature, recovering a result of Wu and Yau.

Motivation & Objective

• To extend the Kähler-Ricci flow approach to the noncompact setting where the holomorphic sectional curvature is bounded above by a negative constant.
• To recover Wu and Yau's result on the existence of Kähler-Einstein metrics with negative scalar curvature on complete noncompact Kähler manifolds with negative holomorphic sectional curvature.
• To avoid reliance on sophisticated tools such as the numerical characterization of the Kähler cone or quasi-bounded coordinate charts used in prior work.
• To establish uniform bounds on curvature and its derivatives using only the maximum principle and parabolic PDE estimates.
• To demonstrate that the normalized Kähler-Ricci flow converges smoothly to a complete Kähler-Einstein metric with negative Ricci curvature.

Proposed method

• Utilizes the normalized Kähler-Ricci flow equation: ∂ₜω(t) = -Ric(ω(t)) - ω(t), starting from an initial metric ω₀ with bounded curvature and negative holomorphic sectional curvature.
• Applies a noncompact version of the maximum principle for parabolic PDEs on complete manifolds with bounded curvature, ensuring suprema are either achieved initially or approached with vanishing gradient and nonnegative time-derivative.
• Establishes a uniform lower bound on the evolving metric ω(t) relative to the initial metric ω₀ via a Schwarz lemma-type estimate adapted to holomorphic sectional curvature.
• Derives uniform bounds on the curvature tensor and its covariant derivatives using an induction argument on the norm of the tensor T = ∇T, leveraging the evolution equation for Sₖ = |∇ᵏT|².
• Uses the evolution of the Kähler potential φ(t) to show C⁰ convergence and then upgrades to C⁴ convergence via uniform Cᵏ bounds on φ, leading to C⁴ convergence of the metrics.
• Applies the maximum principle to the quantity Sₖ = |∇ᵏT|² to derive recursive bounds, ultimately proving uniform Cᵏ bounds on the curvature tensor and its derivatives.

Experimental results

Research questions

• RQ1 Can the normalized Kähler-Ricci flow be used to prove the existence of Kähler-Einstein metrics on complete noncompact Kähler manifolds with negative holomorphic sectional curvature?
• RQ2 Does the flow preserve the negative holomorphic sectional curvature condition over time, and can this be shown without advanced analytic tools?
• RQ3 Can uniform bounds on curvature and its derivatives be established using only the maximum principle and parabolic PDE estimates in the noncompact setting?
• RQ4 Is the normalized Kähler-Ricci flow on such manifolds guaranteed to exist for all time and converge smoothly to a Kähler-Einstein metric?
• RQ5 Can the convergence to a Kähler-Einstein metric be established without relying on the numerical characterization of the Kähler cone or quasi-bounded coordinate charts?

Key findings

• The normalized Kähler-Ricci flow exists for all time on complete Kähler manifolds with bounded curvature and negative holomorphic sectional curvature bounded above by −κ|η|⁴ for some κ > 0.
• The flow converges smoothly to a complete Kähler-Einstein metric ω_KE satisfying Ric(ω_KE) = −ω_KE.
• The limit metric ω_KE is uniformly equivalent to the initial metric ω₀, with C⁻¹ω₀ ≤ ω_KE ≤ Cω₀ for a constant C depending only on B and κ.
• All covariant derivatives of the curvature tensor of ω_KE are uniformly bounded in the ω_KE norm, with |∇ˡ_ω_KE Rm(ω_KE)|_ω_KE ≤ Cₗ for constants Cₗ depending only on B and κ.
• The convergence of the flow is C⁴ smooth, and the Kähler potential φ(t) converges in C⁴ to a limit φ_∞, which defines the limit Kähler-Einstein metric.
• The proof avoids the use of Demailly-Paun’s numerical characterization of the Kähler cone and quasi-bounded coordinate charts, relying instead on elementary maximum principle arguments.