Baker, Charles

### Description

A geometric evolution equation is a partial differential equation that evolves some kind of geometric object in time. The protoype of all parabolic evolution equations is the familiar heat equation. For this reason parabolic geometric evolution equations are also called geometric heat flows or just geometric flows. The heat equation models the physical phenomenon whereby heat diffuses from regions of high temperature to regions of cooler temperature. A defining characteristic of this physical...[Show more] process, as one readily observes from our surrounds, is that it occurs smoothly: A hot cup of coffee left to stand will over a period of minutes smoothly equilibrate to the ambient temperature. In the case of a geometric flow, it is some kind of geometric object that diffuses smoothly down a driving gradient. The most natural extrinsically defined geometric heat flow is the mean curvature flow. This flow evolves regions of curves and surfaces with high curvature to regions of smaller curvature. For example, an ellipse with highly curved, pointed ends evolves to a circle, thus minimising the distribution of curvature. It is precisely this smoothing, energy minimising characteristic that makes geometric flows powerful mathematical tools. From a pure mathematical perspective, this is a useful property because unknown and complicated objects can be smoothly deformed into well-known and easily understood objects. From an applications point of view, it is an observed natural law that physical systems will move towards a state that minimises some notion of energy. As an example, crystal grains will try to arrange themselves so as to minimise the curvature of the interface between them. The study of the mean curvature flow from the perspective of partial differential equations began with Gerhard Huisken's pioneering work in 1984. Since that time, the mean curvature flow of hypersurfaces has been a lively area of study. Although Huisken's seminal paper is now just over twenty-five years old, the study of the mean curvature flow of submanifolds of higher codimension has only recently started to receive attention. The mean curvature flow of submanifolds is the main object of investigation in this thesis, and indeed, the central results we obtain can be considered as high codimension analogues of some early hypersurface theorems. The result of Huisken's 1984 paper roughly says that convex hypersurfaces evolve under the mean curvature flow to round points in finite time. Here we obtain the result that if the ratio of the length of the second fundamental form to the length of the mean curvature vector is bounded (by some explicit constant depending on dimension but not codimension), then the submanifold will evolve under the mean curvature flow to a round point in finite time. We investigate evolutions in flat and curved backgrounds, and explore the singular behaviour of the flows as the first singular time is approached.

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