Homogeneous Einstein manifolds via a cohomogeneity-one approach

Abstract: We establish non-existence results on non-compact homogeneous Einstein manifolds. The key idea in the proof is to consider non-transitive group actions on these spaces (more precisely, actions with cohomogeneity one), and to find geometric monotone quantities for the ODE that results from writing the Einstein equation in such a setting. As an application, we show that homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds. This is joint work with C. Böhm (Münster).