The cosmological constant and higher dimensional dilatation symmetry

We discuss the hypothesis of a fixed point for quantum gravity coupled to a scalar, in the limit where the scalar field goes to infinity, accompanied by a suitable scaling of the metric. We propose that no scalar potential is present for the dilatation symmetric quantum effective action at the fixed point. Dimensional reduction of such a higher dimensional effective action leads to solutions with a vanishing effective four-dimensional constant. Under rather general circumstances these are the only quasistatic stable solutions with finite four-dimensional gravitational constant. If cosmological runaway solutions approach the fixed point as time goes to infinity, the cosmological constant vanishes asymptotically. For our old Universe the fixed point is not yet reached completely, resulting in a tiny amount of dark energy, comparable to dark matter. We discuss explicitly higher dimensional geometries which realize such asymptotic solutions for $t\to\infty$. They include Ricci-flat spaces as well as warped spaces, potentially with singularities.