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

## TL;DR

This paper establishes the first part of a study on the global regularity of a 3+1 dimensional Einstein-Klein-Gordon system with specific symmetries, focusing on reduction to 2+1 dimensions and singularity analysis.

## Contribution

It reduces the Einstein-Klein-Gordon system with $U(1) imes \, \mathbb{R}$ symmetry to a 2+1 dimensional system and analyzes potential singularities.

## Key findings

- Reduction to 2+1 dimensions simplifies the problem.
- Singularities can only occur at the axis.
- Energy estimates and null coordinate system construction are achieved.

## Abstract

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 $U(1)\times \mathbb{R}$ isometry group. In this first part, we reduce the Cauchy problem of the Einstein-Klein-Gordon system to a 2+1 dimensional system. Then, we will give energy estimates and construct the null coordinate system, under which we finally show that the first possible singularity can only occur at the axis.