Global regularity for Einstein-Klein Gordon system with $U(1) \times \mathbb{R}$ isometry group

Abstract

This is an announcement of [3] and [4] which study the global regularity of the 3+1 dimensional Einstein-Klein Gordon system with a $U(1) \times \mathbb{R}$ isometry group. We reduce the Cauchy problem of the Einstein-Klein Gordon system to a 2+1 dimensional system. Next, the energy of this system cannot concentrate near the first possible singularity, and hence is small. Then, we show that the global regularity holds for the reduced 2+1 system with initial data of small energy.