[Paper Review] Some Type I solutions of Ricci flow with rotational symmetry

Jian Song|arXiv (Cornell University)|Mar 13, 2012

Geometry and complex manifolds14 references4 citations

TL;DR

This paper proves that Ricci flow on complex projective space blown up at one point with any rotationally symmetric Kähler metric must develop Type I singularities. Using curvature estimates and Cao's splitting theorem, it shows that under non-collapsing conditions, the parabolic blow-up limit along the exceptional divisor is a complete, non-flat, shrinking gradient Kähler-Ricci soliton on a manifold homeomorphic to ℂⁿ blown up at a point.

ABSTRACT

We prove that the Ricci flow on CP^n blown-up at one point starting with any rotationally symmetric Kahler metric must develop Type I singularities. In particular, if the total volume does not go to zero at the singular time, the parabolic blow-up limit of the Type I Ricci flow along the exceptional divisor is a complete non-flat shrinking gradient Kahler-Ricci soliton on a complete Kahler manifold homeomorphic to C^n blown-up at one point.

Motivation & Objective

• To establish that Ricci flow on ℂℙⁿ blown up at one point with any U(n)-invariant Kähler metric develops Type I singularities.
• To classify the asymptotic behavior of the flow under different volume scaling regimes at the singular time.
• To identify the blow-up limit of the flow along the exceptional divisor when the volume does not collapse.
• To generalize the result to broader classes of Kähler manifolds, including Hirzebruch surfaces and projective bundles over Kähler-Einstein manifolds.

Proposed method

• Apply the unnormalized Ricci flow to Xₙ,ₖ = ℙ(𝒪(ℂℙⁿ⁻¹) ⊕ 𝒪(ℂℙⁿ⁻¹)(-k)) with Gₙ,ₖ-invariant initial Kähler metrics.
• Use the Calabi ansatz to reduce the Kähler-Ricci flow to a system of ODEs on the base ℂℙⁿ⁻¹.
• Establish a uniform lower bound on holomorphic bisectional curvature of the form −C/(T−t) for some C>0.
• Apply Cao’s splitting theorem for Kähler-Ricci flow with nonnegative holomorphic bisectional curvature to rule out Type II singularities.
• Perform parabolic rescaling gⱼ(t′) = Kⱼ g(T + Kⱼ⁻¹ t′) with Kⱼ→∞ to extract the blow-up limit.
• Use Cheeger-Gromov-Hamilton convergence to show the limit is a complete, non-flat, shrinking gradient Kähler-Ricci soliton on a manifold homeomorphic to ℂⁿ blown up at a point.

Experimental results

Research questions

• RQ1 Does Ricci flow on ℂℙⁿ blown up at one point with a U(n)-invariant Kähler metric always develop Type I singularities?
• RQ2 What is the structure of the parabolic blow-up limit along the exceptional divisor when the total volume does not vanish at the singular time?
• RQ3 Can the blow-up limit be identified as a nontrivial complete shrinking gradient Kähler-Ricci soliton?
• RQ4 How does the singularity type depend on the asymptotic behavior of the volume under rescaling?
• RQ5 Can the results be generalized to other Kähler manifolds with similar symmetry, such as Xₙ,ₖ or Xₘ,ₙ,ₖ?

Key findings

• The Ricci flow on ℂℙⁿ blown up at one point with any U(n)-invariant Kähler metric must develop Type I singularities.
• If liminfₜ→ₜ (T−t)⁻¹ Vol(g(t)) = ∞, the rescaled flow subconverges to a complete, non-flat, shrinking gradient Kähler-Ricci soliton on a manifold homeomorphic to ℂⁿ blown up at one point.
• If liminfₜ→ₜ (T−t)⁻¹ Vol(g(t)) ∈ (0,∞), the rescaled flow subconverges to ℂⁿ⁻¹ × ℂℙ¹ with a product metric of flat and Fubini-Study components.
• If liminfₜ→ₜ (T−t)⁻¹ Vol(g(t)) = 0, the rescaled flow converges to the unique compact shrinking Kähler-Ricci soliton on ℂℙⁿ blown up at one point.
• The holomorphic bisectional curvature is uniformly bounded below by −C/(T−t) for some C>0 under non-collapsing conditions.
• The limiting soliton is invariant under a free U(n) action and has the same topological type as ℂⁿ blown up at a point, though biholomorphy is not proven.