Mathematics Colloquium
Eric Ling
University of Copenhagen
Title: Cosmological singularities and the topology of Cauchy surfaces
Date: Monday, January 12
Place and Time: Love 101, 3:05-3:55 pm
Abstract. The celebrated singularity theorems of Hawking and Penrose demonstrate that the existence of singularities, in the sense of timelike or null geodesic incompleteness, is an inevitable feature of generic spacetimes modeling the big bang or black holes. Hawking's original cosmological singularity theorem relied on a curvature condition, called the strong energy condition, which is known to be violated in models with a positive cosmological constant. In this talk, I will show how Penrose's singularity theorem can be adapted to prove a singularity theorem applicable to this regime: if a four-dimensional spacetime satisfying the null energy condition contains a compact Cauchy surface that is expanding in all directions, then the spacetime is past null geodesically incomplete unless the Cauchy surface is topologically a spherical space. The proof makes use of a well-known result in geometric measure theory along with the positive resolutions of the Poincare conjecture and the virtual positive first Betti number conjecture. I will illustrate the theorem with some examples and conclude by analyzing its rigidity under null geodesic completeness.