Ricci Flow Unstable Cell Centered at an Einstein Metric on the Twistor Space of Positive Quaternion Kähler Manifolds of Dimension
نویسندگان
چکیده
We construct a 2-parameter family FZ of Riemannian metrics on the twistor space Z of a positive quaternion Kähler manifold M satisfying the following properties : (1) the family FZ contains an Einstein metric gZ and its scalings, (2) the family FZ is closed under the operation of making the convex sums, (3) the Ricci map g 7→ Ric(g) defines a dynamical system on the family FZ, (4) the Ricci flow starting at any metric in the family FZ stays in FZ and is an ancient solution having the Einstein metric.gZ as its asymptotic soliton. This means that the family FZ is a 2-dimensional “unstable cell” w.r.to the Ricci flow which is “centered” at the Einstein metric gZ. We apply the estimates for the covariant derivative of the curvature tensor under the Ricci flow to this “unstable cell” and settle the LeBrun-Salamon conjecture : any irreducible positive quaternion Kähler manifold is isometric to one of the Wolf spaces.
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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 Ri...
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