Séance Séminaire

Séminaire Gaston Darboux

Einstein 4-manifolds, negative curvature and smoothing cones

I will describe joint work with Bruno Premoselli which gives a new existence theorem for negatively curved Einstein 4-manioflds, which are obtained by ?smoothing? the singularities of hyperbolic cone metrics. Let (M_k) be a sequence of compact 4-manifolds and let g_k be a hyperbolic cone metric on M_k with cone angle \alpha (independent of k) along a smooth surface S_k. We make the following assumptions: 1. The normal injectivity radii i(k) of S_k tends to infinity. 2 A point at distance least i(k)/2 from S_k has injectivity radius at least i(k)/2 . 3. The area of the singular locii satisfy A(S_k)\leq C \exp(5 i(k)/2) for some C independent of k. When these assumptions hold, we prove that for all large k, M_k carries a smooth Einstein metric of negative curvature. The proof involves a gluing theorem and a parameter dependent implicit function theorem (where k is the parameter). As I will explain, negative curvature plays an essential role in the proof. (For those who may be aware of our arxiv preprint, https://arxiv.org/abs/1802.00608, the work I will describe has a new feature, namely we now treat all cone angles, and not just those which are greater than 2\pi. This gives lots more examples of Einstein 4-manifolds.)