Rigidity and flexibility of anti-de Sitter 3-manifolds

Topology is the mathematical theory of shapes. At the beginning of the twentieth century, early topologists quickly learnt to describe all possible two-dimensional shapes. However, when turning to three-dimensional shapes, they encountered serious problems. In the 1970s, William Thurston discovered that most three- dimensional shapes admit a so-called hyperbolic geometric structure. Hyperbolic geometry behaves quite differently from what we are familiar with in Euclidean geometry (and not much three-dimensional shapes admit a Euclidean geometric structure). This discovery was a big surprise even to the mathematical community of that time. Hyperbolic geometry has since become a very active area of research. It has lead to important insights and has helped to answer many questions. On the other hand, in the theory of general relativity, we consider so-called Lorentzian spaces, where one dimension is timelike and behaves very differently from the other, spacial dimensions. Anti-de Sitter geometry is the Lorentzian analogue of hyperbolic geometry. Over the last few decades, Anti-de Sitter geometry has played an important role in building new physical theories, as well as in looking at some classical mathematical objects in a completely new way. In my project, I will investigate three-dimensional anti-de Sitter spaces. I will explore how such spaces are determined by boundary data and their behavior at geometric singularities. My research will be guided by analogies with hyperbolic geometry.

The project was focused on the geometry of (2+1)-dimensional spacetimes. These are useful for models of quantum gravity, for developing the toolbox of general relativity theory, and for mathematical understanding of possible geometric structures. The (2+1)-spacetimes that are relevant for physics have constant curvature. The main achievement of the project is a proof that under some conditions such spacetimes are uniquely determined only by the intrinsic geometry of two its spacelike slices. Some possibilities of the geometry of the slices were described. Similar results were obtained for hyperbolic 3-manifolds, which resemble constant-curvature (2+1)-spacetimes in some of their properties, and which are foundational in 3-dimensional topology. Some applications to other mathematical topics were outlined.