UC Berkeley

First, we show that a Ricci flow can be started from a non-compact complete manifold, if the manifold is non-collapsed and satisfies a lower bound for many known curvature conditions.In this theorem, we do not need the manifold to have bounded curvature, which was assumed in an earlier work by Bamler-Cabezas-Rivas-Wilking.

Second, we show that a Ricci flow can be started from a 3d complete manifold with non-negative Ricci curvature.This gave a partial affirmative answer to a conjecture by Topping. we prove it by generalizing the concept of singular Ricci flow by Kleiner and Lott to non-compact initial conditions.

Third, we find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies Hamilton's flying wing conjecture. We also show that the scalar curvature does not vanish at infinity in a 3d flying wing. For dimension $n\ge 4$, we find a family of non-collapsed, $\mathbb{Z}_2\times O(n-1)$-symmetric, but non-rotationally symmetric n-dimensional steady gradient solitons with positive curvature operator.

This thesis is a composition of the following three papers of the author: "Ricci flow under local almost non-negative curvature conditions", "Producing 3d Ricci flows with non-negative Ricci curvature via singular Ricci flows", "A family of 3d steady gradient solitons that are flying wings" \cite{Lai2019,Lai2020,Lai2020a}.