Members & Former Members

For every \(n \geq 4\) we construct infinitely many mutually not homotopic closed manifolds of dimension \(n\) which admit a negatively curved Einstein metric but no locally symmetric metric.

Related project(s):
51The geometry of locally symmetric manifolds via natural maps

Extending earlier work of Tian, we show that if a manifold admits a metric that is almost hyperbolic in a suitable sense, then there exists an Einstein metric that is close to the given metric in the \(C^{2,\alpha}\)-topology. In dimension 3 the original manifold only needs to have finite volume, and the volume can be arbitrarily large. Applications include a new proof of the hyperbolization of 3-manifolds of large Hempel distance yielding some new geometric control on the hyperbolic metric, and an analytic proof of Dehn filling and drilling that allows the filling and drilling of arbitrary many cusps and tubes.

Related project(s):
51The geometry of locally symmetric manifolds via natural maps