Dynamic instability of $\mathbb{CP}^N$ under Ricci flow

TL;DR

This paper provides an independent proof that complex projective spaces with the Fubini--Study metric are dynamically unstable under Ricci flow in all complex dimensions greater than one, challenging existing conjectures about stability.

Contribution

It offers a new proof of the instability of $ ext{CP}^N$ under Ricci flow and highlights that the instability is not limited to Kähler perturbations, countering previous conjectures.

Findings

$ ext{CP}^N$ is dynamically unstable under Ricci flow for all $N>1$

The instability perturbation is not Kähler

Challenges the conjecture on the stability of Kähler metrics under Ricci flow

Abstract

The intent of this short note is to provide context for and an independent proof of the discovery of Klaus Kroencke that complex projective space with its canonical Fubini--Study metric is dynamically unstable under Ricci flow in all complex dimensions N>1. The unstable perturbation is not Kaehler. This provides a counterexample to a well known conjecture widely attributed to Hamilton. Moreover, it shows that the expected stability of the subspace of Kaehler metrics under Ricci flow, another conjecture believed by several experts, needs to be interpreted in a more nuanced way than some may have expected.