## Publicly Accessible Penn Dissertations

2022

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

Wolfgang Ziller

Renato G. Bettiol

#### Abstract

We investigate the behaviour of the Ricci flow for homogeneous metrics on spheres and on general compact homogeneous spaces. In particular we complete the classification of ancient homogeneous solutions on spheres and discover a new 1-parameter family of ancient solutions. These solutions can be described in terms of shrinking the fibers of the Hopf fibration $S^1\to S^{4n+3}\to \mathbb{CP}^{2n+1}$ while varying the metric on the $\mathbb{CP}^{2n+1}$ base. Precisely one solution collapses along the backwards flow to the Fubini-Study metric while the rest collapse to Ziller's second Einstein metric on $\mathbb{CP}^{2n+1}$. We then proceed to determine a general criterion for the existence of collapsed ancient solutions on compact homogeneous spaces. In particular, we show that whenever $G/H$ is the total space of a homogeneous fibration $T^n\to G/H\to G/K$ where $T^n$ is a maximal torus in a compact complement of $H$ in $N_G(H)$, then for every Einstein metric on the base $G/K$ there exists a family of ancient solutions on $G/H$ which collapse to the given Einstein metric under the backwards flow. This construction generalizes all previously known examples of collapsed homogeneous ancient solutions in the literature, and also leads to many new families of examples.

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