On the Ricci Flow of Homogeneous Metrics on Spheres

2020/06/03 by Sbiti, Sammy
#53C30 #53C44 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2006.02566

Abstract

We study the Ricci flow of the four-parameter family of Sp(n+1)-invariant metrics on spheres. We determine their forward behaviour and also classify ancient solutions. In doing so, we exhibit a new one-parameter family of ancient solutions on spheres. These (non-isometric) ancient solutions all have a larger isometry group, namely Sp(n+1)Sp(1), Sp(n+1)U(1), or U(2n+2). Two ancient solutions are non-collapsed and converge, under the backwards flow, to Jensen's second Einstein metric. One solution parametrizes the well known Berger metrics. The rest are new and collapse, under a rescaling of the backwards flow, to Ziller's second homogeneous Einstein metric on complex projective space.

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