Ricci Flow on CP1-bundles over a Product of Kähler-Einstein Manifolds

Technical Summary: Ricci Flow on $\mathbb{CP}^1$-Bundles over a Product of Kähler-Einstein Manifolds

Problem Statement
This paper investigates the behavior of the Ricci flow on a specific class of manifolds: $\mathbb{CP}^1$-bundles over a product of compact Kähler-Einstein manifolds, $N = N_1 \times \cdots \times N_r$. The study focuses on initial metrics $g_0$ constructed via a specific geometric ansatz originally developed by Wang-Wang [WW98] and Dancer-Wang [DW11] for constructing Einstein metrics and Ricci solitons. The primary objectives are twofold: first, to determine if this specific metric ansatz is preserved under the Ricci flow evolution; and second, to classify the nature of finite-time singularities that may arise, specifically distinguishing between Type I and Type II singularities in the Kähler case.

Methodology
The authors employ a combination of geometric analysis, ODE reduction, and curvature estimation techniques:

1. Geometric Ansatz and Structure: The manifold $M$ is modeled as a submersion $\pi: P \to N$ with circle fibers, extended to a manifold $M \cong I \times P$. The metric is defined by a warped product structure involving a radial parameter $s$, a contact 1-form $\eta$, and conformal factors $H(s)$ and $F_i(s)$ scaling the base metrics $g_{N_i}$. The authors rigorously define the closing conditions required for the metric to be smooth and complete at the endpoints of the interval $I$.
2. Curvature Computation: Utilizing O'Neill's tensors for Riemannian submersions, the authors compute the Ricci curvature tensor of the ansatz metric. They demonstrate that the Ricci tensor retains the same structural form as the metric itself (a linear combination of $ds^2$, $\eta \otimes \eta$, and the pullbacks of the base metrics).
3. Reduction to ODEs: By establishing that the Ricci tensor shares the metric's structure, the authors prove that the Ricci flow evolution reduces to a system of coupled parabolic partial differential equations (PDEs) for the conformal factors. These PDEs are further shown to be equivalent to a system of ODEs under the flow, ensuring the ansatz is preserved.
4. Singularity Analysis:
   • Preservation: The preservation of the ansatz is proven via the existence and uniqueness of solutions to the derived ODE system.
   • Gradient Estimates: The authors apply Li-Yau's gradient estimates to the heat-type equations satisfied by the squared conformal factors $F_i^2$. This yields uniform bounds on the gradients of these factors.
   • Schwarz-type Estimates: By analyzing the behavior of the Laplacian of $F_i^2$ at the boundary of the manifold, the authors derive a linear lower bound for the conformal factors in terms of the extinction time $T$, specifically $F_i^2 \geq C(T-t)$.
   • Blow-up Argument: To classify the singularity, the authors assume a Type II singularity and perform a parabolic rescaling. They utilize the derived gradient bounds and the specific curvature decomposition (involving O'Neill's $A$-tensor) to show that the rescaled limit would split isometrically into a product involving a flat factor and a 2-dimensional steady Ricci soliton (a "cigar"). They argue this contradicts Perelman's local non-collapsing theorem, thereby ruling out Type II singularities.

Key Contributions and Results

• Preservation of Ansatz: The paper proves (Proposition 3.1) that the specific metric ansatz used by Wang-Wang and Dancer-Wang is preserved under the Ricci flow. Consequently, the evolution of the metric is governed entirely by the evolution of the scalar functions $H(s,t)$ and $F_i(s,t)$.
• Type I Singularity Classification: The main result (Theorem 4.5) establishes that for the Kähler-Ricci flow on these manifolds, any finite-time singularity must be of Type I. This means the curvature blows up at a rate bounded by $C/(T-t)$.
• Generalization of Symmetry: The work extends previous results by Fong [Fon14], Song [Son15], and Guo-Song [GS16], which were restricted to base manifolds that were single Kähler-Einstein manifolds with Calabi's $U(n)$-symmetry. This paper generalizes the setting to allow the base to be a product of Kähler-Einstein manifolds, thereby relaxing the symmetry requirements on the base while maintaining the fiber structure.
• Curvature Decomposition: The authors provide explicit formulas for the Riemann curvature tensor in terms of the base curvatures and the conformal factors, utilizing O'Neill's tensors to handle the interaction between the fiber and the product base.

Significance
The paper claims significance by enriching the understanding of Ricci flow singularities on manifolds with symmetry. While previous works focused on single base factors or higher-rank bundles over single bases, this work addresses the complexity introduced by a product base structure. The authors note that their result confirms the likelihood of Type I singularities occurring when the manifold possesses "good symmetry" (in this case, the specific bundle structure).

The authors are modest regarding the identification of the specific blow-up models for all cases. They confirm that if the singularity is caused by the collapsing of the $\mathbb{CP}^1$-fibers, the blow-up limit is the product $\mathbb{CP}^1 \times \mathbb{C}^n$ (where $n = \dim_{\mathbb{C}} N$), consistent with prior findings. However, they explicitly state that if the singularity arises from the contraction of specific factors of the base manifold (but not all), the nature of the blow-up limit is "not straight-forward" and remains an open question to be addressed in forthcoming work. Thus, the paper's primary contribution is the rigorous proof of the Type I nature of the singularity and the preservation of the ansatz, rather than a complete classification of all possible blow-up models.