Month: April 2022

29 April 2022

Globally hyperbolic spacetimes: slicings, boundaries and counterexamples

The Cauchy slicings for globally hyperbolic spacetimes and their relation with the causal boundary are surveyed and revisited. Our study covers: (1) adaptative possibilities and techniques for their Cauchy slicings, (2) global hyperbolicity of sliced spacetimes, and (3) computation of the causal boundary of a Cauchy temporal splitting. Based on arXiv:2110.13672.

22 April 2022

Poincaré-Einstein metrics with cusps

Asymptotically hyperbolic (or Poincaré-Einstein) Einstein metrics are important in conformal geometry and the physics of the AdS-CFT correspondence.  In 1991, Graham and Lee proved that any sufficiently small perturbation (in Holder norm) of the round metric on the unit sphere (with dimension at least 3) is the “conformal infinity” of an asymptotically hyperbolic Einstein metric on the open unit ball.  In this talk I will give the background and review the proof of this result.  I then discuss recent joint work with Rochon proving the existence of Einstein metrics near certain geometrically finite quotients of the hyperbolic metric.

8 April 2022

Regge-Teitelboim conditions (and their relationship with harmonic coordinates)

In General Relativity, an “isolated system at a given instant of time” is modeled as an asymptotically Euclidean initial data set (M,g,K). Such asymptotically Euclidean initial data sets (M,g,K) are characterized by the existence of asymptotic coordinates in which the Riemannian metric g and second fundamental form K decay to the Euclidean metric \delta and to 0 suitably fast, respectively. Using harmonic coordinates Bartnik showed that (under suitable integrability conditions on their matter densities) the (ADM-)energy, (ADM-)linear momentum and (ADM-)mass of an asymptotically Euclidean initial data set are well-defined. To study the (ADM-)angular momentum and (BORT-)center of mass, however, one usually assumes the existence of Regge-Teitelboim coordinates on the initial data set (M,g,K) in question, i.e. the existence of asymptotically Euclidean coordinates satisfying additional decay assumptions on the odd part of g and the even part of K. We will show that, under certain circumstances, harmonic coordinates can be used as a tool in checking whether a given asymptotically Euclidean initial data set possesses Regge-Teitelboim coordinates. This allows us to easily give examples of (vacuum) asymptotically Euclidean initial data sets which do not possess any Regge-Teitelboim coordinates. This is joint work with Carla Cederbaum and Jan Metzger.

1 April 2022

Geometry and Topology: A Tale of Rigidity

A mainstream in mathematics is to study the relation between the geometry and topology of a manifold. The geometry is about the distance, length, area, volume and determined by its curvature. The topology is about properties that are preserved under continuous deformations. It is generally observed that a suitable assumption on the geometry would lead to certain restrictions on the topology. In this talk, we’ll discuss some recent progress in this direction, particularly the joint work with Xiaodong Cao and Matthew Gursky leading to a resolution of a conjecture proposed by S. Nishikawa.