Ricci Flow Unstable Cell Centered at a Kähler-einstein Metric on the Twistor Space of Positive Quaternion

We show that there exists a 2-parameter family F of Riemannian metrics on the twistor space Z of a positive quaternion Kähler manifold M having the following properties : (1) the family F is closed under the operation of making the convex sums, (2) the Ricci map g 7→ Ric(g) sends the family F to itself, (3) the family F contains the scalings of a Kähler-Einstein metric of Z. We show that the Ricci flow starting at any metric in the family F stays in F and is an ancient solution whose asymptotic solution corresponds to the KählerEinstein metric. This means that the family F is a 2-dimensional unstable cell w.r.to the Ricci flow which is “centered” at the trajectory consisting of the scalings of a positive Kähler-Einstein metric. We combine this fact with the estimate for the covariant derivative of the curvature tensor under the Ricci flow and settle the LeBrun-Salamon conjecture claiming that any irreducible positive quaternion Kähler manifold is isometric to one of the Wolf spaces. §