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

Frederick Tsz-Ho Fong, Hung Tran

TL;DR

We study the Kähler-Ricci flow on a $\mathbb{CP}^1$-bundle over a product of compact Kähler-Einstein manifolds using a Wang-DW type ansatz with equidistant hypersurfaces. The ansatz is shown to be preserved under the flow and, in the Kähler setting, every finite-time singularity is Type I; the analysis combines O'Neill tensor calculus, Li-Yau gradient estimates, and Schwarz-type bounds to control curvature and fiber metrics. The results extend Calabi symmetry-type results to products of KE bases and elucidate possible blow-up models, including fiber-collapse limits and divisor-contraction limits, via blow-up and Cheeger-Gromov arguments. This framework advances understanding of singularity formation in symmetric Ricci flows on fiber bundles and guides the identification of blow-up limits.

Abstract

In this paper, we study the Ricci flow on CP1-bundles over a product of Kähler-Einstein manifolds whose initial metric is constructed by the ansatz used in works by M. Wang et. al. We prove that the ansatz is preserved along the Ricci flow. Furthermore, in the Kähler case, we proved that Type I finite-time singularity must occur under such an ansatz.

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

TL;DR

We study the Kähler-Ricci flow on a -bundle over a product of compact Kähler-Einstein manifolds using a Wang-DW type ansatz with equidistant hypersurfaces. The ansatz is shown to be preserved under the flow and, in the Kähler setting, every finite-time singularity is Type I; the analysis combines O'Neill tensor calculus, Li-Yau gradient estimates, and Schwarz-type bounds to control curvature and fiber metrics. The results extend Calabi symmetry-type results to products of KE bases and elucidate possible blow-up models, including fiber-collapse limits and divisor-contraction limits, via blow-up and Cheeger-Gromov arguments. This framework advances understanding of singularity formation in symmetric Ricci flows on fiber bundles and guides the identification of blow-up limits.

Abstract

In this paper, we study the Ricci flow on CP1-bundles over a product of Kähler-Einstein manifolds whose initial metric is constructed by the ansatz used in works by M. Wang et. al. We prove that the ansatz is preserved along the Ricci flow. Furthermore, in the Kähler case, we proved that Type I finite-time singularity must occur under such an ansatz.
Paper Structure (15 sections, 10 theorems, 132 equations)

This paper contains 15 sections, 10 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. An Ansatz with Equidistant Hypersurfaces
  3. The ansatz
  4. Closing conditions
  5. Special case: Calabi symmetry
  6. O'Neill Tensors
  7. Computation of Curvature
  8. Reduction of Ricci flow to ODEs
  9. Type I Singularity
  10. Riemann curvature in the Kähler case
  11. Li-Yau's gradient estimates
  12. Schwarz's estimates
  13. Type I singularity
  14. Discussion on Blow-up Models of Singularity
  15. Appendix

Key Result

Lemma 2.1

For the ansatz eq:DW, the $A$-tensor is given by for any $X, Y \in \mathcal{H}$ and $U, V \in \mathcal{V}$.

Theorems & Definitions (23)

  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 3.1
  • ...and 13 more