Area-constrained Willmore surfaces in initial data sets

Which surface is the roundest? At first, this question might appear trivial: It is, of course, the surface of a ball. However, the problem already becomes a lot more challenging when the surface is required to have a hole. In fact, it has only been ten years ago that the optimal surface was found to resemble the surface of an inflated swim ring. The quest for such particularly round surfaces becomes even more interesting once we leave the well-known Euclidean space. In general relativity, the universe we observe is described as a four-dimensional curved spacetime (a surprising consequence of this curvature is the fact that time does not elapse at the same pace everywhere in space). It turns out that this spacetime can be understood fully if one has precise knowledge about a snapshot of space at a fixed time. Contrary to the Euclidean space, such a three-dimensional slice of spacetime is a curved space. The round surfaces contained in this slice encode important information on the distribution of matter in our spacetime. In many situations, it is not known if a curved space contains such round surfaces, let alone, what properties these surfaces have. The goal of my research is to find out how many of these round surfaces exist in the three-dimensional slices that occur in general relativity and to obtain a better understanding of the relationship between the geometry of these surfaces and the physical properties of our spacetime.

In general relativity, the universe we observe is described as a four-dimensional curved spacetime. It turns out that this spacetime can be understood fully if one has precise knowledge about a snapshot of space at a fixed time. Contrary to the Euclidean space, such a three-dimensional slice of spacetime is a curved space. So-called area-constrained Willmore surfaces, which are particularly round surfaces, encode important information on the distribution of matter in such a slice of our spacetime. Within this project, it has been shown that a large part of such a slice can be foliated by area-constrained Willmore surfaces, which are essentially unique. The positioning of these surfaces has been shown to measure the distribution of matter in a very precise way. What is more, the techniques developed within this project have found many other applications in geometry and mathematical relativity.