# The stability of Kerr-de Sitter black holes (Part 2)

In this lecture I will discuss Kerr-de Sitter black holes, which are rotating black holes in a universe with a positive cosmological constant, i.e. they are solutions (in 3+1 dimensions) of Einstein's equation $$\mathrm{Ric}(g)+\Lambda g=0$$ with $\Lambda>0$ constant, and $g$ a Lorentzian metric, of signature $(1,3)$, given by an explicit formula, due to Kerr and Carter. They are parameterized by their mass and angular momentum, much like Kerr black holes in a space-time with vanishing cosmological constant.

I will discuss the geometry of these black holes as well as that of the underlying de Sitter space, and then talk about the stability question for these black holes in the initial value formulation. Namely, appropriately interpreted (fixing a gauge), Einstein's equations can be thought of as quasilinear wave equations, and then the question is if perturbations of the initial data along a Cauchy hypersurface produce solutions which are close to, and indeed asymptotic to, a Kerr-de Sitter black hole, typically with a different mass and angular momentum. In this lecture, which is a continuation of my Colloquium lecture, I will discuss the analysis (partial differential equations) aspects of the Kerr-de Sitter black hole stability problem. The key aspect is analysis of linear wave equations on asymptotically Kerr-de Sitter backgrounds, including with coefficients possessing limited regularity. This talk is also based on joint work with Peter Hintz.