Ricci Flow on CP1-bundles over a Product of K\"ahler-Einstein Manifolds

TL;DR

This paper investigates the behavior of Ricci flow on CP1-bundles over products of K"ahler-Einstein manifolds, demonstrating the preservation of a specific metric ansatz and the inevitability of Type I singularities in the K"ahler case.

Contribution

It proves the preservation of a particular metric ansatz under Ricci flow and establishes finite-time singularity formation in the K"ahler case for these bundles.

Findings

The ansatz remains intact during Ricci flow.

Type I finite-time singularities occur in the K"ahler case.

The results extend understanding of Ricci flow on complex bundles.

Abstract

In this paper, we study the Ricci flow on CP1-bundles over a product of K\"ahler-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 K\"ahler case, we proved that Type I finite-time singularity must occur under such an ansatz.