[Paper Review] Pseudolocality for the Ricci flow and applications

Albert Chau, Luen-Fai Tam+1 more|ArXiv.org|Jan 5, 2007

Geometric Analysis and Curvature Flows15 references17 citations

TL;DR

This paper establishes pseudolocality and Li-Yau-Hamilton (LYH) type inequalities for the Ricci flow on complete non-compact Riemannian manifolds with bounded curvature and vanishing curvature at infinity. Using these estimates, it proves that finite-time singularities must be confined to compact sets, and extends long-time existence results for the Kähler-Ricci flow on complete non-compact Kähler manifolds with non-negative holomorphic bisectional curvature and asymptotically flat geometry.

ABSTRACT

In \cite{P1}, Perelman established a differential Li-Yau-Hamilton (LYH) type inequality for fundamental solutions of the conjugate heat equation corresponding to the Ricci flow on compact manifolds (also see \cite{N2}). As an application of the LYH inequality, Perelman proved a pseudolocality result for the Ricci flow on compact manifolds. In this article we provide the details for the proofs of these results in the case of a complete non-compact Riemannian manifold. Using these results we prove that under certain conditions, a finite time singularity of the Ricci flow must form within a compact set. We also prove a long time existence result for the \KRF flow on complete non-negatively curved \K manifolds.

Motivation & Objective

• To extend Perelman’s pseudolocality and Li-Yau-Hamilton (LYH) inequality results from compact to complete non-compact Riemannian manifolds.
• To establish conditions under which finite-time singularities in the Ricci flow must be confined to compact subsets.
• To prove long-time existence of the Kähler-Ricci flow on complete non-compact Kähler manifolds with non-negative holomorphic bisectional curvature and vanishing curvature at infinity.
• To generalize existing results on curvature decay and flow extension using gradient estimates and fundamental solution bounds.

Proposed method

• Derives gradient estimates and fundamental solution bounds for the conjugate heat equation associated with the Ricci flow on complete non-compact manifolds.
• Establishes a differential LYH inequality for the fundamental solution of the conjugate heat equation under bounded curvature and asymptotic flatness.
• Applies the LYH inequality to prove pseudolocality: regions of high curvature cannot instantly influence almost Euclidean regions.
• Uses pseudolocality to show that if the Ricci flow develops a finite-time singularity, curvature must blow up only on a compact set.
• Combines curvature decay at infinity with volume growth and scalar curvature integral estimates to control the Kähler-Ricci flow's long-time behavior.
• Lifts the Kähler-Ricci flow to the universal cover and uses volume growth and scalar curvature decay to bound the logarithm of the volume ratio, ensuring uniform control.

Experimental results

Research questions

• RQ1 Can Perelman’s pseudolocality and LYH inequality results for the Ricci flow be extended to complete non-compact Riemannian manifolds?
• RQ2 Under what conditions does a finite-time singularity in the Ricci flow on a non-compact manifold occur only within a compact set?
• RQ3 Does the Kähler-Ricci flow on a complete non-compact Kähler manifold with non-negative holomorphic bisectional curvature and vanishing curvature at infinity admit a long-time solution?
• RQ4 Can curvature decay at infinity and volume growth estimates be used to control the logarithm of the volume ratio and ensure long-time existence?
• RQ5 What role does the universal cover and holomorphic splitting play in proving long-time existence of the Kähler-Ricci flow?

Key findings

• The pseudolocality theorem holds for complete non-compact manifolds: if the Ricci flow develops a finite-time singularity, curvature blow-up is confined to a compact set.
• Under the assumption that |Rm|(x) → 0 as x → ∞ and injectivity radius is bounded away from zero, the Ricci flow either exists for all time or singularities are compactly supported.
• If T < ∞ in Theorem 1.1, then Rm(x,t) → 0 as x → ∞ uniformly in t ∈ [0,T), meaning curvature decays at infinity uniformly over time.
• For complete non-compact Kähler manifolds with non-negative holomorphic bisectional curvature and |Rm|(x) → 0 as x → ∞, the Kähler-Ricci flow exists for all time t ∈ [0, ∞).
• The logarithm of the volume ratio F(x,t) = log(det g(x,t)/det g(x,0)) is uniformly bounded above on M × [0,T), which implies curvature is uniformly bounded and allows extension beyond T.
• The proof relies on volume growth estimates and scalar curvature decay on the universal cover, showing that −F(x,t) is bounded via integral estimates involving ∫s/(1+s) ds, which controls the growth of the volume ratio.