Date of Award

2019

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Graduate Group

Mathematics

First Advisor

Wolfgang Ziller

Abstract

In the first part of this thesis, in joint work with Renato Bettiol, we show that the geometric property of nonnegative sectional curvature is not preserved under the Ricci flow on closed manifolds of dimension greater than or equal to 4. This is in contrast to the situation for 3 dimensional manifolds. The main strategy is to study the Ricci flow equation on certain 4 dimensional manifolds that admit an isometric group action of cohomogeneity one.

Along the way we need to show that a certain canonical form for an invariant metric on a cohomogeneity one manifold, is preserved under the Ricci flow. In the particular situation of the above mentioned result, we prove the preservation of that canonical form using an ad hoc method. It is an interesting question whether this canonical form for a cohomogeneity one metric is preserved in general. In the second

part of the thesis we present a strategy to tackle this problem, explain its geometric consequences, and also explain the challenges in carrying out the strategy, along with some partial results.

Included in

Mathematics Commons

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