# Joel Fine, Université Libre de Bruxelles: "Examples of compact Einstein 4-manifolds with negative curvature"

Nov 07, 2018 from 12:15 PM to 01:45 PM

LMS 4 - Raum 423 - Konferenzraum

Abstract:

I will describe joint work with Bruno Premoselli, which finds new compact Einstein 4-manifolds. They are seemingly the first examples with negative curvature which are not locally symmetric. The starting point is a construction of Gromov and Thurston, of a sequence of hyperbolic n-manifolds M_k each of which contains a totally geodesic codimension 2 submanifold S_k, which is null-homologous. Moreover the normal injectivity radius of S_k goes to infinity with k. Let X_k be the d-fold cover of M_k branched along S_k. We prove that when n=4 and for all large k, X_k carries an Einstein metric of negative curvature. The first step is to find an approximate solution to Einstein’s equations on X_k. (This works for any n.) The second step is to prove that there is a genuine solution nearby. This is done with a parameter dependent implicit function theorem. The analysis is surprisingly delicate and, so far at least, only works when n=4.