The branching curves and their application to the four dimensional Ricci flow

TL;DR

This paper introduces a novel approach using branching curves and local invariants to analyze four-dimensional Ricci flow, providing new insights into singularity formation and models.

Contribution

It develops a new framework based on branching curves to study Ricci flow in four dimensions, reformulating key theorems and characterizing singularity models.

Findings

Reformulation of the Cheeger-Gromov-Hamilton Compactness Theorem using branching curves.

Characterization of Type I singularity models in four-dimensional Ricci flow.

New local invariants for analyzing Ricci flow singularities.

Abstract

We study the four dimensional Ricci flow with the help of local invariants. If $(M^4, g(t))$ is a solution to the Ricci flow and $x \in M^4$, we can associate to the point $x$ a one-parameter family of curves, which lie in the product of two projective lines. This allows us to reformulate the Cheeger-Gromov-Hamilton Compactness Theorem in the context of these curves. We use this result, in order to study Type I singularities in dimension four and give a characterization of the corresponding singularity models.