# On 3+1 Lorentzian Einstein Manifolds with One Rotational Isometry

## TL;DR

This paper studies 3+1 Lorentzian Einstein manifolds with rotational symmetry and positive cosmological constant, reducing the problem to 2+1 equations and proving positive mass-energy theorems without smallness assumptions.

## Contribution

It introduces a reduction of 3+1 Einstein equations with symmetry to 2+1 equations coupled with wave maps and establishes explicit positive mass-energy theorems.

## Key findings

- Reduction to 2+1 Einstein equations with wave maps
- Explicit positive mass-energy theorems proved
- No smallness assumptions required

## Abstract

We consider 3+1 rotationally symmetric Lorentzian Einstein spacetime manifolds with $\Lambda >0$ and reduce the equations to 2+1 Einstein equations coupled to `shifted' wave maps. Subsequently, we prove various (explicit) positive mass-energy theorems. No smallness is assumed.