Global Regularity for Einstein-Klein-Gordon System with U(1)×R Isometry Group,I

This is the first of the two papers devoted to the study of global regularity of the 3+1 dimensional Einstein-Klein-Gordon system with a U(1)×R isometry group.In this first part,the authors reduce the Cauchy problem of the Einstein-Klein-Gordon system to a 2+1 dimensional system.Then,the authors will give energy estimates and construct the null coordinate system,under which the authors finally show that the first possible singularity can only occur at the axis.