The canonical Einstein metric on $G_{2}$ is dynamically unstable under the Ricci flow

TL;DR

This paper proves that the bi-invariant Einstein metric on the compact Lie group G2 is dynamically unstable under Ricci flow, completing the stability analysis for such metrics on compact simple Lie groups and highlighting G2's uniqueness among exceptional groups.

Contribution

It demonstrates the instability of the bi-invariant Einstein metric on G2 under Ricci flow, filling a gap in the stability classification of metrics on compact simple Lie groups.

Findings

G2's Einstein metric is dynamically unstable under Ricci flow.

Completes the stability analysis for bi-invariant metrics on compact simple Lie groups.

G2 is uniquely unstable among exceptional groups.

Abstract

In this note we show that the bi-invariant Einstein metric on the compact Lie group $G_{2}$ is dynamically unstable as a fixed point of the Ricci flow. This completes the stability analysis for the bi-invariant metrics on the compact, connected, simple Lie groups. Interestingly, $G_{2}$ is the only unstable exceptional group.