Ricci Flow Singularity for Triaxial Bianchi IX Metric

TL;DR

This paper investigates conditions under which a Type I singularity develops in Ricci flow on 4D manifolds with triaxial Bianchi IX metrics, extending previous symmetric cases.

Contribution

It generalizes prior work by establishing sufficient conditions for singularity formation in less symmetric, triaxial Bianchi IX geometries under Ricci flow.

Findings

Identifies conditions leading to Type I singularities.

Extends previous symmetric manifold results.

Provides a framework for analyzing singularities in more general geometries.

Abstract

In this work, we are going to find sufficient conditions on the initial triaxial Bianchi IX metric on some 4-dimensional manifolds foliated by homogeneous S3 for a Type I singularity to occur when it is flowed under the Ricci flow. This work generalises the study on rotationally symmetric manifolds done by Angenent and Isenberg as well as the work of Isenberg, Knopf, and Sesum, in which they introduced some ansatz for the problem setup.