We introduce a class of overdetermined systems of partial differential equations of on (pseudo)-Riemannian manifolds that we call the generalised Ricci soliton equations. These equations depend on three real parameters. For special values of the parameters they specialise to various important classes of equations in differential geometry. Among them there are: the Ricci soliton equations, the vacuum near-horizon geometry equations in general relativity, special cases of Einstein-Weyl equations and their projective counterparts, equations for homotheties and Killing's equation. We provide explicit examples of generalised Ricci solitons in 2 dimensions, some of them obtained using techniques developed by J. Jezierski. This is joint work with Pawel Nurowski available at arXiv:1409.4179.

In this talk, we will discuss asymptotically hyperbolic initial data for the Einstein equations modeling asymptotically null slices in asymptotically Minkowski spacetimes. Such initial data consists of a Riemannian manifold $(M,g)$ whose geometry at infinity approaches that of hyperbolic space, and a symmetric 2-tensor $K$ representing the second fundamental form of the embedding into spacetime, such that $K$ approaches $g$ at infinity. Just like in the asymptotically Euclidean setting, positive mass conjecture for asymptotically hyperbolic initial data can be proven by spin techniques in all dimensions. However, without spin assumption only partial results are available, even in the important particular case $K=g$, where the conjecture merely states that an asymptotically hyperbolic manifold whose scalar curvature is greater than or equal to the scalar curvature of hyperbolic space must have positive mass unless it is hyperbolic space. Having reviewed the available results, we will present a non-spinor proof of positive mass theorem for asymptotically hyperbolic initial data sets in dimension 3. The argument uses the Jang equation to reduce the proof to the application of the celebrated Riemannian positive mass theorem for asymptotically Euclidean manifolds and can potentially be extended to all dimensions less than 8.