K\"ahler-Ricci flow and conformal submersion

TL;DR

This paper investigates the behavior of the K"ahler-Ricci flow on manifolds with a special conformal submersion, identifying conditions for its preservation and singularity formation, extending previous symmetry-based results.

Contribution

It provides necessary and sufficient conditions for preserving conformal submersions under the flow and describes singularity formation, generalizing Calabi symmetry results.

Findings

Conditions for preservation of conformal submersion derived

Type I singularity formation established

Standard splitting of the limit shown

Abstract

We study singularity formation of K\"ahler-Ricci flow on a K\"ahler manifold that admits a horizontally homothetic conformal submersion into another K\"ahler manifold. We will derive necessary and sufficient conditions for the preservation of horizontally homothetic conformal submersion along the flow and establish the formation of type I singularity together with a standard splitting of the Cheeger-Gromov limit. This generalizes the setup of Calabi symmetry that was discussed in \cite{1} and \cite{2}, thus gives new proofs for the results listed there.