[Paper Review] Diameter and Ricci curvature estimates for long-time solutions of the Kahler-Ricci flow

Wangjian Jian, Jian Song|arXiv (Cornell University)|Jan 12, 2021

Geometry and complex manifolds45 references5 citations

TL;DR

This paper establishes uniform diameter and Ricci curvature bounds for long-time solutions of the normalized Kähler-Ricci flow on Kähler manifolds with semi-ample canonical bundle. Under the assumption that the canonical bundle is semi-ample, the authors prove that the diameter remains uniformly bounded and Ricci curvature is uniformly bounded away from singular fibers, extending Perelman-type estimates to the long-time regime in the minimal model program framework.

ABSTRACT

It is well known that the Kähler-Ricci flow on a Kähler manifold $X$ admits a long-time solution if and only if $X$ is a minimal model, i.e., the canonical line bundle $K_X$ is nef. The abundance conjecture in algebraic geometry predicts that $K_X$ must be semi-ample when $X$ is a projective minimal model. We prove that if $K_X$ is semi-ample, then the diameter is uniformly bounded for long-time solutions of the normalized Kähler-Ricci flow. Our diameter estimate combined with the scalar curvature estimate in [34] for long-time solutions of the Kähler-Ricci flow are natural extensions of Perelman's diameter and scalar curvature estimates for short-time solutions on Fano manifolds. We further prove that along the normalized Kähler-Ricci flow, the Ricci curvature is uniformly bounded away from singular fibres of $X$ over its unique algebraic canonical model $X_{can}$ if the Kodaira dimension of $X$ is one. As an application, the normalized Kähler-Ricci flow on a minimal threefold $X$ always converges sequentially in Gromov-Hausdorff topology to a compact metric space homeomorphic to its canonical model $X_{can}$, with uniformly bounded Ricci curvature away from the critical set of the pluricanonical map from $X$ to $X_{can}$.

Motivation & Objective

• To extend Perelman's short-time diameter and scalar curvature estimates to long-time solutions of the Kähler-Ricci flow on minimal models.
• To establish uniform diameter bounds under the assumption that the canonical bundle is semi-ample.
• To prove uniform Ricci curvature bounds away from singular fibers in the case of Kodaira dimension one.
• To analyze the Gromov-Hausdorff limit of the normalized Kähler-Ricci flow on minimal threefolds.
• To confirm convergence of the flow to the canonical model as a compact metric space with controlled curvature.

Proposed method

• Use of the normalized Kähler-Ricci flow equation with initial Kähler metric $ g_0 $, defined as $ \partial_t g = -\text{Ric}(g) - g $.
• Application of the pluricanonical map $ \Phi: X \to X_{\text{can}} $ to define the open set $ X^\circ = \Phi^{-1}(X_{\text{can}}^\circ) $, where $ X_{\text{can}}^\circ $ is the Zariski open set over which fibers are nonsingular.
• Construction of a Lyapunov-type function $ K = \rho^l H $, where $ \rho $ is a cutoff function and $ H $ is related to the normalized potential $ u - \bar{u} $, to control curvature quantities.
• Employment of a maximum principle argument on $ K $ to derive uniform upper bounds on the scalar curvature and Ricci curvature components.
• Use of the twisted Kähler-Einstein metric $ g_{\text{can}} $ on $ X_{\text{can}}^\circ $ satisfying $ \text{Ric}(g_{\text{can}}) = -g_{\text{can}} + g_{WP} $, where $ g_{WP} $ is the Weil-Petersson metric.
• Establishment of uniform bounds on $ |\text{Ric}| $ on compact subsets $ \mathcal{K} \subset X^\circ $ via pointwise estimates on Ricci curvature components and scalar curvature control.

Experimental results

Research questions

• RQ1 Does the diameter of the Kähler manifold remain uniformly bounded along the normalized Kähler-Ricci flow when the canonical bundle is semi-ample?
• RQ2 Is the Ricci curvature uniformly bounded away from the singular fibers of the pluricanonical map when the Kodaira dimension is one?
• RQ3 Can the long-time solution of the normalized Kähler-Ricci flow on a minimal threefold converge in Gromov-Hausdorff topology to its canonical model?
• RQ4 To what extent do Perelman-type estimates for diameter and scalar curvature extend to long-time solutions in the minimal model program?
• RQ5 How does the curvature behavior near the singular fibers of the pluricanonical fibration affect the limiting metric structure?

Key findings

• The diameter of the Kähler manifold is uniformly bounded for all time along the normalized Kähler-Ricci flow when the canonical bundle $ K_X $ is semi-ample.
• For manifolds of Kodaira dimension one, the Ricci curvature is uniformly bounded away from the singular fibers of the pluricanonical map.
• On a minimal threefold, the normalized Kähler-Ricci flow converges sequentially in Gromov-Hausdorff topology to a compact metric space homeomorphic to the canonical model $ X_{\text{can}} $.
• The Ricci curvature remains uniformly bounded on any compact subset $ \mathcal{K} \subset X^\circ $, the regular locus of the pluricanonical fibration.
• The limiting metric space is the metric completion of $ (X_{\text{can}}^\circ, g_{\text{can}}) $, which is compact and homeomorphic to $ X_{\text{can}} $.
• The proof relies on a maximum principle applied to a carefully constructed function $ K = \rho^l H $, yielding uniform bounds on curvature quantities via pointwise estimates.