Ancient Solutions to Null Mean Curvature Flow

In general relativity, objects can move through space at most with the speed of light. In particular, information about far away celestial objects can not reach us instantly, but rather travels as radiation that has been moving through space for thousands of years. The collection of all lightrays emanating off of a given source, like a star, forms a so called lightcone in the mathematical model. Lightcone, since lightrays in the model spread out from a point at an angle of 45 degrees. In my project, I want to understand how surfaces along such a lightcone deform under a geometric flow. Along a geometric flow, surfaces deform according to their curvature at each point, and the process ends in a surface of constant curvature. A good example of such a constant curvature surface is a circle which remains unchanged under a geometric flow. On the other hand, a shape that has been bent into an oval shape will deform back into a perfect circle. In this way, geometric flows are a powerful tool to find surfaces with special properties. A particularly interesting case is the study of ancient solutions to such a flow, that is solutions which exists for all times to the past and which is a very strong assumption. Again, the circle is a good example as it always has been a circle in its past (and will remain to be one). Circles moreover play a role in motivating the center of mass general relativity from a geometric perspective. Here, Einsteins idea that the physical properties of a space are explained by how it is curved plays a key role. To illustrate this, consider a piece of cloth that has been drawn taut and an iron ball which bends the cloth under its weight. Without the ball the cloth lays flat and we may place a circular piece of wire anywhere on the cloth. However, if we put the ball back, it is easy to see that the only way to neatly place the wire on the cloth is to place it centered around the ball. In this way, one can identify the center of mass of the ball with the center of the wire. In a more complicated model, one can use more abstract concepts such as constant curvature surfaces or ancient solutions to a geometric flow to simulate the properties of a circle and still motivate a notion of center of mass in a geometric way. With this motivation at hand, I aim to translate these concepts to the case of a lightcone in my project. The lightcone case is of particular interest as information from far away galaxies reaches us along these objects. Therefore, theoretical definitions of center of mass could potentially be compared to real physical measurements.