Finite Time Singularities of Collapsing Kähler Ricci Flow on Ruled Surfaces

Abstract

We prove that any finite time collapsing Kähler Ricci flow on ruled surfaces develops a Type I singularity, such singularity is modeled on the standard product shrinker $\mathbb{P}^1\times \mathbb{C}$. As an application, we obtain the optimal collapse rate of fibers on ruled surfaces.