Evolutions of $S^3$ and $\mathbb{R}P^3$ that describe Eguchi-Hanson metric and metrics of constant curvature

TL;DR

This paper explores the evolution of 3-spheres under Ricci flow and introduces new flows that describe 4-dimensional metrics of constant curvature and the Eguchi-Hanson metric, linking geometric flows with special Einstein metrics.

Contribution

It introduces a novel Dirac flow to describe 4D constant curvature metrics and a new flow leading to the Eguchi-Hanson metric, expanding the understanding of geometric evolutions.

Findings

Describes Ricci flow evolution of $S^3$

Introduces Dirac flow for constant curvature metrics

Defines a new flow leading to Eguchi-Hanson metric

Abstract

In this work we illustrate some well-known facts about the evolution of $S^3$ under the Ricci flow. The Dirac flow we introduce allows us to describe the 4- dimensional metrics with constant curvature. Another new flow leads to the Eguchi-Hanson metric and can be defined either on metric or on corresponding contact forms.