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

## TL;DR

This paper proves the global regularity of solutions for a 2+1 dimensional Einstein-Klein-Gordon system with specific symmetries, extending previous work on singularity formation and energy concentration.

## Contribution

It establishes global existence for small energy initial data in a reduced Einstein-Klein-Gordon system with U(1) × R symmetry, building on prior reduction to 2+1 dimensions.

## Key findings

- Energy does not concentrate near the first potential singularity.
- Global regularity holds for initial data with small energy.
- Singularity can only occur at the axis in the reduced system.

## Abstract

This paper is devoted to the study of the global existence of smooth solutions for the 3+1 dimensional Einstein-Klein-Gordon systems with a $U(1) \times \mathbb{R}$ isometry group for a class of regular Cauchy data. In our first paper \cite{chen}, we reduce the Einstein equations to a 2+1 dimensional Einstein-wave-Klein-Gordon system. And we show that the first possible singularity can only occur at the axis. In this paper, we give a proof for the global regularity for the 2+1 dimensional system. Firstly, we show the non-concentration of the energy near the first possible singularity. Then, we prove that the global regularity holds for initial data with small energy.