# On the linear stability of nearly-K\"ahler $6$-manifolds

## TL;DR

This paper proves that strict nearly K"ahler 6-manifolds with nonzero second or third Betti number are linearly and dynamically unstable under Ricci flow, using Perelman's $
u$-entropy.

## Contribution

It establishes the instability of certain nearly K"ahler 6-manifolds under Ricci flow based on topological Betti numbers.

## Key findings

- Strict nearly K"ahler 6-manifolds with nonzero Betti numbers are linearly unstable.
- Such manifolds are also dynamically unstable under Ricci flow.
- The instability is demonstrated via Perelman's $
u$-entropy analysis.

## Abstract

We show that a strict, nearly K\"ahler $6$-manifold with either second or third Betti number nonzero is linearly unstable with respect to the $\nu$-entropy of Perelman and hence is dynamically unstable for the Ricci flow.