Doctoral Dissertations

Date of Award

8-2024

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Theodora Bourni, Mathew Langford

Committee Members

Vasileios Maroulas, George Siopsis

Abstract

Following the tremendous success of the mean curvature flow, other variants such as the Gauss curvature flow, inverse mean curvature flow have been investigated in great detail, leading to interesting applications to other fields including partial differential equations, convex geometry etc. This calls for an investigation of curvature flow as a general phenomenon. While basic existence and uniqueness results, roundness estimates etc have been obtained, there isn't a substantial body of work that addresses the geometry of solutions of curvature flows and their relation to the choice of speed function used. It is therefore interesting to investigate curvature flows as a general phenomenon from a more abstract and axiomatic point of view.

In this volume, we take such an abstract point of view, and study a certain important class of solutions called \emph{ancient solutions}; in particular, we study translating solutions and the so-called ``pancake" solutions. We provide a detailed construction of these solutions, and prove their uniqueness as well. We also provide a description of their geometry. The advantage of such an abstract approach is that we can see the dependence of the solution's geometry on basic properties of the speed function, often just algebraic properties.

Finally, we also take a discrete point of view to curvature, tackling some questions about the curvature of a network (i.e. graph) and its relation to the network topology.

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